Laurent Series Rings and Related Rings 9783110702163, 9783110702248, 9783110702309, 2020940691

Table of contents :
Contents
Introduction
1 Preliminary properties of A((x, φ)) and M((x, φ))
2 Noetherian rings A((x, φ))
3 Serial and Bezout rings A((x, φ))
4 Prime and semiprime skew Laurent series rings
5 Regular and biregular Laurent series rings
6 Equivalent definitions of Laurent rings
7 Generalized Laurent rings
8 Properties of Laurent rings
9 Laurent rings: examples, relation
10 Noetherian and Artinian Laurent rings
11 Simple and semisimple Laurent rings
12 Uniserial and serial Laurent rings
13 Semilocal Laurent rings
14 Filtrations and (generalized) Malcev–Neumann rings
15 Properties of generalized Malcev–Neumann rings
16 Properties and examples of Malcev–Neumann rings
17 Laurent series in two variables
Bibliography
Notation
Index

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Askar Tuganbaev Laurent Series Rings and Related Rings

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Askar Tuganbaev

Laurent Series Rings and Related Rings |

Mathematics Subject Classification 2010 16S99, 16S36, 16S32 Author Prof. Dr. Askar Tuganbaev National Research University MPEI Department of higher mathematics Krasnokazarmennaya 14 Moscow 111250 Russian Federation [email protected]

ISBN 978-3-11-070216-3 e-ISBN (PDF) 978-3-11-070224-8 e-ISBN (EPUB) 978-3-11-070230-9 Library of Congress Control Number: 2020940691 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Cover image: emer1940 / iStock / Getty Images Plus Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

|

Dedicated to the memory of Diar Tuganbaev

Contents Introduction | XI 1

Preliminary properties of A((x, φ)) and M((x, φ)) | 1

2

Noetherian rings A((x, φ)) | 10

3

Serial and Bezout rings A((x, φ)) | 17

4

Prime and semiprime skew Laurent series rings | 31

5

Regular and biregular Laurent series rings | 39

6

Equivalent definitions of Laurent rings | 47

7

Generalized Laurent rings | 53

8

Properties of Laurent rings | 63

9

Laurent rings: examples, relation | 67

10

Noetherian and Artinian Laurent rings | 81

11

Simple and semisimple Laurent rings | 87

12

Uniserial and serial Laurent rings | 91

13

Semilocal Laurent rings | 98

14

Filtrations and (generalized) Malcev–Neumann rings | 106

15

Properties of generalized Malcev–Neumann rings | 113

16

Properties and examples of Malcev–Neumann rings | 118

17

Laurent series in two variables | 126

VIII | Contents Bibliography | 129 Notation | 133 Index | 135

Introduction In the book, we study properties of skew Laurent series rings A((x, φ)) over a ring A, where A is an associative (not necessarily commutative) ring with non-zero identity element; A is called the coefficient ring of the ring A((x, φ)). In the text, we use base ring-theoretical information which can be found, for example, in [9, 10] and [21]. The use of skew Laurent series rings began in papers of Schur, Dickson and Hilbert at the beginning of the 20th century. For example, in the study of the independence of axioms in geometry, Hilbert used skew Laurent series rings to construct a division ring which is infinite-dimensional over its center. The study of Laurent series rings with arbitrary coefficient ring began in papers [29, 47] and [51]. The use of Laurent series rings is useful in the ring theory. For example, it is shown in [30], with the use of skew Laurent series rings in two variables, that the ring of fractions of the Weil algebra contains a free non-commutative subalgebra. In [14], Laurent series rings are used for the study of Krull dimension and global dimension of Noetherian PI rings. In addition, the ring-theoretical properties of Laurent series rings are studied, for example, in papers [52, 53, 54, 55, 59, 60, 61, 62, 63, 64, 65, 66, 67, 70]. We denote by A+ ((x)) the Abelian group consisting of formal series f = ∑n∈ℤ fn x n such that all its coefficients fn are contained in the ring A and fn = 0 for almost all negative integers n; the element f0 of the ring A is called the constant term of the series. The elements fn ∈ A are said to be canonical coefficients of the series f . In the group A+ ((x)), addition and subtraction are defined by the relation f ± g = ∑n∈ℤ (fn ± gn )x n , the series with zero coefficients is the zero element in A+ ((x)). We denote by A+ [[x]] the subgroup in A+ ((x)) consisting of series f such that fn = 0 for all negative inten gers n. It is clear that every non-zero series f in A+ ((x)) has the form f = ∑+∞ n=k(f )∈ℤ fn x , k(f ) 0 ≠ fk(f ) ∈ A, and it can be uniquely represented in the form f = f ̄(x)x , where f ̄(x) ∈ A[[x]] and the series f ̄(x) has the non-zero constant term f ̄ = f ; the non-zero 0

k(f )

element fk(f ) of the ring A is called the lowest coefficient of the series f , the product of fk xk is called the lowest term of the series f , the integer k(f ) is called the lowest degree of the series f . We assume that the lowest degree of the zero series is equal to +∞. Let A be a ring with automorphism φ. The first main aim of this book is an exposition of ring-theoretical properties of the (left) skew Laurent series ring A((x, φ)) over the ring A. The additive group of the ring A((x, φ)) is the group A+ ((x)) and the product fg of two series f , g ∈ A((x, φ)) is naturally defined, considering the relation xn a = φn (a)x n . In the ring A((x, φ)), the additive subgroup A+ [[x]] is a subring which is called the (left) skew power series ring; it is denoted by A[[x, φ]]. If φ = 1A is the identity automorphism of the ring A, then we write A((x)) and A[[x]] instead of A((x, 1A )) and A[[x, 1A ]], respectively, i. e., the additive groups A+ ((x)) and A+ [[x]] turn into rings which are called the (ordinary) Laurent series ring and the (ordinary) power series ring over the ring A, respectively.

https://doi.org/10.1515/9783110702248-201

X | Introduction Ring-theoretical properties of the rings A((x, φ)) and A[[x, φ]] have many differences. For example, the power series ring A[[x, φ]] always has the non-zero Jacobson radical J(A[[x, φ]]) containing the non-zero ideal consisting of series with zero constant terms, and the Jacobson radical J(A((x, φ))) of the Laurent series ring A((x, φ)) is often equal to the zero (for example, if A is a ring without non-zero nilpotent elements or a semiprime, right or left Goldie ring; see Lemma 2.6(7) and Lemma 1.8(4), although the Jacobson radical of the coefficient ring A can be non-zero; see, for example, A = F[[x]], where F is a field and J(A((x))) = 0. We note also that, if φ is the automorphism of complex conjugation the complex number field ℂ, then ℂ((x, φ)) is a non-commutative division ring (see 1.2(6)). The pseudo-differential operator algebra A((t −1 , δ)) was defined by Schur in [49] and since then repeatedly used in various sections of mathematics (e. g., see [11, 48, 32]). In this book, the ring-theoretical properties of pseudo-differential operator rings are only studied. Therefore, we do not mention papers on pseudo-differential operators not related to structural ring theory. We only mention the papers [11] and [34]. In the latter paper, the author develops an algebraical theory of formal pseudodifferential operator rings in several variables and notes that “other ways to construct pseudo-differential operator rings are given in [16, 17, 5]”. In the same paper, iterated skew Laurent series rings are used. In structural ring theory, pseudo-differential operator rings are used for calculations in differential operator algebras (see [13]) and for construction various examples (see, for example, [15]). If a pseudo-differential operator ring has the right Krull dimension, then it is a right Noetherian ring [55]. The ring properties of the rings A((x, φ)) and A((t −1 , δ)) are close to each other. In these rings, a variable does not commute with coefficients and the difference between the rings is only that a relation acts as a substitute for commutativity: xa = φ(a)x or ta = at + δ(a). In the case of the identity automorphism φ and the zero derivation δ, these rings are isomorphic to the ordinary Laurent series ring A((x)). It follows from the above that it is convenient to define the class of Laurent rings which properly includes all skew Laurent series rings and all pseudo-differential operator rings. This notion was defined in [69]. All results of the book on Laurent rings are based on the paper [69]. Many results on skew Laurent series rings and pseudodifferential operator rings are obtained as corollaries of similar results for Laurent rings. We also construct other examples of Laurent rings, for which also many results of the book are true. In this book, the required computing technique in Laurent rings is developed. The book also contains various examples of Laurent rings and the required definitions and notation. The skew Laurent series ring A((x, φ)) and the pseudo-differential operator ring A((t −1 , δ)) over the same ring A are isomorphic to each other as Abelian groups and multiplication in them are similarly defined. Many theorems can be transferred from skew Laurent series rings to pseudo-differential operator rings, and conversely, almost unchanged. Therefore, the following question appears: what should be the multipli-

Introduction

| XI

cation on the additive group of formal series in order to maintain the same ring properties. It is clear that there must be natural conditions resulting from the distributivity of the multiplication with respect to the formal infinite sum and from the identification the unit of the coefficient ring with the unit of the cyclic group of multiplication generated by the variable. It turns out that the unique additional required property consists of in the property that lowest degree product of two series cannot be less than the sum of lowest degrees of these series (in the case of the pseudo-differential operator ring, where the degree of the variable in the formal series decreases, instead of the lowest degrees are used the leading degrees). Rings satisfying this condition and consisting of formal power series with a finite number of negative degrees of a variable are called Laurent rings in this book. By considering the fact that x is invertible, we obtain from this condition that we have the required condition for the transposition relation xa = ⋅ ⋅ ⋅ which consists of the property that the lowest degree of the right part has to be equal to lowest degree of the left part. Since x is invertible, the relation has to be of the form xa = φ(a)x + ⋅ ⋅ ⋅ , where φ is an automorphism of the coefficient ring. The associativity of multiplication imposes the last condition on the relation, which will be more convenient for us to formulate after the development of a special computing technique. Verification of this condition is difficult in the general case, but in some special cases it can be easily verified. Thus, a ring of skew Laurent series with skew differentiation is constructed (it was studied in [4] in the case when the coefficient ring is a division ring), which also turns out to be a special case of a Laurent ring. There is a certain relationship between the lattice of right (resp., left) ideals of the Laurent ring and the lattice of right (resp., left) ideals of its coefficient ring: using the map μ, the lattice of right (resp., left) ideals of the coefficient ring is embedded (with preservation lattice operations) into the lattice of right (resp., left) of the Laurent ring and there exists a map λ in the opposite direction, preserving the inclusion relation, associating with each right (resp., left) ideal of the ring of series the right (resp., left) ideal of the coefficient ring. In the finitely generated case, the map λ also preserves a strict inclusion, so that the lattice of right (resp., left) ideals of Laurent series cannot be much more sophisticated than the lattice of right (resp., left) ideals of its coefficient ring. After obtaining the basic results, we study the specific ring properties of Laurent rings (all these results naturally extend to skew Laurent series rings and pseudodifferential operator rings). So, it is proved that a Laurent ring is a division ring if and only if its coefficient ring is a division ring; for partial cases of skew Laurent series rings and pseudo-differential operator rings, this fact is well known (for example, see [4, with 66] and [13]). Similarly, a Laurent ring is Noetherian (resp., Artinian) ring if and only if its coefficient ring is a Noetherian (resp., Artinian) ring; this assertion is known for skew Laurent series rings and pseudo-differential operator rings (for example, see [15, 55, with 19]). It is also proved that a Laurent ring is a domain if and

XII | Introduction only if its coefficient ring is a domain. It is proved that a Laurent ring is a principal right ideal domain if and only if its coefficient ring is a principal right ideal domain. A criterion is obtained for a Laurent ring to be a uniserial ring and a criterion to be a distributive semilocal ring. In these cases, the coefficient ring and the Laurent ring are Artinian rings. There is also obtained a description of serial skew Laurent series rings. If a Laurent ring is a simple (resp., semisimple)1 ring, then the coefficient ring is a simple (resp., semisimple) ring and some additional condition holds (this is a condition for the twisting automorphism in the case of the skew Laurent series ring and a condition for the derivation in the case of the pseudo-differential operator ring). In addition to addressing exact criteria, some partial results were obtained on distributive series rings, on semilocal series rings, and on rings of principal right ideals. For many statements, examples of rings are given illustrating the need for each individual condition. Laurent series rings are closely related to Malcev–Neumann series rings and generalized power series rings, which have been intensively studied recently. We recall that Malcev–Neumann series rings were defined in 1948 by Malcev to prove the embeddability of a group algebra over a field in a division ring (independently, this construction was defined by Neumann in 1949). Among the many papers in this direction, we note [3, 29, 47, 51] and [33]. Rings of generalized power series with exponents in an ordered monoid have been studied by many authors [38, 39, 40, 41, 42, 43, 44, 45, 46, 6, 23, 24, 25, 26, 27, 28]. In [70], Malcev–Neumann rings are defined. The class of all Malcev–Neumann rings properly contains Malcev–Neumann series rings, skew formal Laurent series rings and pseudo-differential operator rings. In [70], the ring-theoretical properties of Malcev–Neumann rings are studied. It turns out that Malcev–Neumann series rings, skew formal Laurent series rings and formal pseudo-differential operator rings have close ring-theoretical properties associated with the existence of the filtrations with respect to the lowest degrees of series. We give some definitions and notation used in the book. Let A be a ring and M a right A-module. A ring is said to be regular if for every its element a, there exists an element x with axa = a. A ring is regular if and only if every its principal right ideal is generated by an idempotent. A ring is said to be reduced if it does not contain non-zero nilpotent elements. A module is said to be Rickartian if all its cyclic submodules are projective. A right module over the ring A is a Rickartian if and only if the annihilator of every its element is generated by an idempotent of the ring A as right ideal. A ring is said to be Rickartian right (left) if it is a Rickartian right (left) module over itself. 1 A semisimple ring is an Artinian semisimple ring.

Introduction

| XIII

A module is said to be uniserial if all its submodules form a chain with respect to inclusion. A module is said to be serial if it is the direct sum of uniserial modules. A ring A is said to be a principal right ideal ring if every its right ideal is a principal right ideal. A module M is called a Bezout module if every of its finitely generated submodules is cyclic. A ring A is said to be a right Bezout ring if every of its finitely generated right ideals is a main, i. e., A is a right Bezout A-module. It is clear that every homomorphic image of the Bezout module (of right Bezout ring) is a Bezout module (right Bezout ring). A module is said to be uniform if any two of its non-zero submodules have a nonzero intersection, i. e., every its non-zero submodule is an essential2 submodule. A module is said to be finite-dimensional if it does not contain an infinite direct sum of non-zero submodules. A module is said to be quotient finite-dimensional if all its factor modules are finite-dimensional. A finitely generated module M is said to be local (resp., semilocal) if its factor module M/J(M) modulo the Jacobson radical J(M) is a simple (resp., semisimple) module, i. e., if J(M) is a maximal submodule in M (resp., J(M) is the intersection of a finite number of maximal submodules of the module M). We denote by max M the set of all maximal submodules of the module M. A ring A is a local (resp., semilocal) ring if and only if its factor ring modulo the Jacobson radical J(A) is a division ring (semisimple Artinian ring). A ring is said to be semiprime if B = 0 for any of its right ideal B with B2 = 0. A ring is said to be prime (resp., a domain) if the product of any two of its non-zero right ideals (resp., elements) is non-zero. A proper ideal B of the ring A is said to be semiprime (resp., prime; completely prime) if the factor ring A/B is semiprime (resp., prime; a domain). A ring A is said to be semiprimary if its the Jacobson radical J(A) is nilpotent and the factor ring modulo the Jacobson radical is a semisimple ring. The intersection of all prime ideals of the ring A is called the prime radical of the ring A. A right finite-dimensional ring with maximum condition on right annihilators is called a right Goldie ring. For a ring A, the group of invertible elements is denoted by U(A). For a module M, a submodule N of M is said to be essential if for any submodule X of M, the relation X ∩ N = 0 implies the relation X = 0. We denote by Sing M the fully invariant submodule of the module M consisting of all elements m of the module M such that the annihilator of the element m is an essential right ideal of the ring A; Sing M is called the singular submodule of M. If Sing M = 0, then the module M is said to be nonsingular. The ideal Sing AA of the ring A is called the right singular ideal. The intersection of all maximal submodules of the module M is denoted by J(M) and is 2 For the module M its submodule X is said to be essential if X ∩ Y ≠ 0 for every non-zero submodule Y of the module M.

XIV | Introduction called the Jacobson radical of the module M. The endomorphism ring of the module M is denoted by End M. A module M is said to be semiprimitive if J(M) = 0. A ring A is said to be semilocal if A/J(A) is a semisimple Artinian ring. A ring A is said to be semiprimary if A is a semilocal ring and the ideal J(A) is nilpotent.

1 Preliminary properties of A((x, φ)) and M((x, φ)) 1.1 The left power series ring A[[x, φ]], where φ is an injective (not necessarily surjective) endomorphism of the ring A Let φ be an injective endomorphism of the ring A. On the additive group A+ [[x]] of left power series with coefficient ring A, we can naturally define multiplication such that xn a = φn (a)xn for all a ∈ A and any positive integer n. As a result, we obtain a ring left power series A[[x, φ]] which in the general case has distinct properties as a right A[[x, φ]]-module and a left A[[x, φ]]-module. This ring consists of sei ries f = ∑+∞ i=0 fi x , where fi ∈ A for all non-negative integers i and x is a formal variable. i 1. For any series g ∈ A[[x, φ]], the series ∑+∞ i=0 (gx) ∈ A[[x, φ]] is defined correctly, since we have to summarize only a finite number of elements of the ring A to calculate the coefficients of this series. 2. A ring A is a domain if and only if the ring A[[x, φ]] is a domain. i 3. For any series g ∈ A[[x, φ]], the series 1 + ∑+∞ i=0 (gx) ∈ A[[x, φ]] is well defined. This series is invertible in the ring A[[x, φ]] and the series 1 − gx is the inverse series for i the series 1 + ∑+∞ i=0 (gx) , since the relations +∞

+∞

i=0

i=0

(1 − gx)(1 + ∑ (gx)i ) = 1 = (1 + ∑ (gx)i )(1 − gx) are directly verified. 4. If a is an element of the ring A which is right invertible in the ring A[[x, φ]], then the element a is right invertible in the ring A. 5. If a is a right invertible element of the ring A, then, for any series g ∈ A[[x, φ]], the series a + gx is right invertible in the ring A[[x, φ]]. ◁ 1, 2, 3. The assertions are directly verified. 4. If af = 1 and f ∈ A[[x, φ]], then af0 = 1. 5. Let ab = 1, where b ∈ A. By 3, the series 1 + bgx is invertible in the ring A[[x, φ]]. Since a + gx = a(1 + bgx), we have (a + gx)(1 + bgx)−1 b = a(1 + bgx)(1 + bgx)−1 b = ab = 1. ▷

1.2 Skew Laurent series rings and modules Let A be a ring with automorphism φ. We recall that A((x, φ)) is the left skew Laurent i series ring consisting of formal series ∑∞ i=t ai x in the variable x with canonical coefhttps://doi.org/10.1515/9783110702248-001

2 | 1 Preliminary properties of A((x, φ)) and M((x, φ)) ficients ai ∈ A, where t is an integer (maybe negative) and either at ≠ 0 or ai = 0 for all i. In A((x, φ)), addition is naturally defined and multiplication is defined with the use of the relations xi a = φi (a)xi for all a ∈ A and i ∈ ℤ. i The elements ai ∈ A are called canonical coefficients of the series f = ∑∞ i=t ai x . If at ≠ 0, then the non-zero coefficient at ∈ A is called the lowest coefficient for f ; it is denoted by λ(f ); if f = 0, then λ(f ) = 0 by definition. If f ≠ 0, then the element at xt and the integer t are called the lowest term and the lowest degree of the series f , respectively. For every subset F in A((x, φ)), we denote by λ(F) the subset {λ(f ) | f ∈ i F} ⊆ A. For every subset B ⊆ A, we denote by B((x, φ)) the subset {∑∞ i=t bi x | bi ∈ B} in A((x, φ)). Similarly, if φ is an automorphism of the ring A, then the right skew Laurent sei ries ring ((φ, x))A = Ar ((x, φ)) consists of formal series ∑∞ i=t x ai in the variable x with i i i canonical coefficients ai ∈ A, where t ∈ ℤ and ax = x φ (a) for all a ∈ A and i ∈ ℤ. The power series rings A[[x, φ]] and [[φ, x]]A are subrings of the rings A((x, φ)) and ((φ, x))A, respectively. We recall also that, for every right A-module M, we denote by M((x, φ)) the set of i all formal series f = ∑∞ i=t mi x , where mi ∈ M, m ∈ ℤ, and either mt ≠ 0 or mi = 0 for all i. The non-zero coefficient mt ∈ M is denoted by λ(f ); it is called the lowest coefficient of the series f . (By definition, we assume that λ(0) = 0.) If f ≠ 0, then the element mt xt and the integer t are called the lowest term and the lowest degree of the series f , respectively. The set M((x, φ)) is a right A((x, φ))-module, where module addition is defined naturally and multiplication by elements of A((x, φ)) is defined by the relation ∞

∞

i=t

j=s

∞

(∑ mi xi )(∑ aj xj ) = ∑ ( ∑ mi φi (aj ))x k . k=t+s i+j=k

For every subset F in M((x, φ)), we denote by λ(F) the subset {λ(f ) | f ∈ F} ⊆ A. If F is an A((x, φ))-submodule in M((x, φ)), then λ(F) is a submodule in MA . For every subset N in MA , we denote by N((x, φ)) the subset of the right A((x, φ))-module M((x, φ)) consisting of all series whose coefficients are contained in N. Therefore, the ring A((x, φ)) is a right A((x, φ))-module of Laurent series with coefficients in AA . Similarly, we define the right A[[x, φ]]-module M[[x, φ]] of skew power series and the right A[x, x−1 , φ]-module M[x, x−1 , φ] of skew Laurent polynomials with coefficients in the module M. For any subset N of the module M, we denote by N((x, φ)) the subset of the i A((x, φ))-module M((x, φ)) consisting of series f = ∑+∞ i=k fi x such that all coefficients fi are contained in set N. Similarly, we consider the subset N[[x, φ]] of the ring A[[x, φ]].

1.2 Skew Laurent series rings and modules | 3

1.

2.

3.

4.

5. 6. 7.

8.

If P is a submodule of the Laurent series module M((x, φ)) and λ(P) is the set of lowest coefficients of all series in P, then λ(P) is a submodule of the module M. If Q is submodule of a right A-module M and Q is the set of all series f ∈M((x, φ)) such that all coefficients are contained in the module Q, then the set Q is a submodule of the right A((x, φ))-module M((x, φ)) and the mapping Q → Q is an injective lattice homomorphism from the submodule lattice of the module M into the A((x, φ))-submodule lattice of the module M((x, φ)). We can consider the ring A (and the ring A((x, φ))) as a right module over itself. Then the notations λ(B) and B are passed to right ideals of these rings. In addition for any ideal B of the ring A((x, φ)), the right ideal λ(B) of the ring A turns out to be an ideal and λ(B) = φ(λ(B)) Every non-zero series f in M((x, φ)) can be uniquely represented in the form f = gxk , where g is a non-zero series in M[[x, φ]] and k ∈ ℤ. In addition S = {1, x, x2 , x3 , . . .} is a multiplicatively closed Öre subset in A[[x, φ]] and the ring A((x, φ)) coincides with ring of fractions of the ring A[[x, φ]] with respect to S. If the φ is not the identity automorphism, then the ring A((x, φ)) is not commutative. A ring A is a domain if and only if the ring A((x, φ)) is a domain. If f ∈ A((x, φ)) and the lowest coefficient ft of the series f is right invertible (resp., left) in the ring A, then the series f is right invertible (resp., left) in the ring A((x, φ)). In addition, if the series f is contained in A[[x, φ]], then every its right (resp., left) inverse the element is contained in A[[x, φ]]. Consequently, the ring A((x, φ)) is a division ring if and only if A is a division ring. If φ is an automorphism of complex conjugation the field of complex numbers ℂ, then ℂ((x, φ)) is a non-commutative division ring.

◁ 1–4. The assertions are directly verified. 5. If a ∈ A and a ≠ φ(a), then ax ≠ xa = φ(a)x. 6. The assertion is verified with the use of 4 and 1.1(2). 7. Let A((x, φ)) be a division ring and 0 ≠ a ∈ A. Since A((x, φ)) is a division ring, af = fa = 1 for some series f ∈ A((x, φ)). Then af0 = f0 a = 1, where f0 is the constant term of the series f . Therefore, A is a division ring. Now let A be a division ring and 0 ≠ f = gxk ∈ A((x, φ)), where k ∈ ℤ, g ∈ A[[x, φ]], 0 ≠ g0 ∈ A. Since the non-zero element g0 is invertible in the division ring A, then the series g is invertible in the ring A[[x, φ]], by 1.1(5). In particular, the series g is invertible in the ring g ∈ A((x, φ)). Then the product f = gx k of two invertible elements of the ring A((x, φ)) is an invertible element, and A((x, φ)) is a division ring. 8. The assertion follows from 5 and 7. ▷

4 | 1 Preliminary properties of A((x, φ)) and M((x, φ))

1.3 The ring A((x, φ)) with inner automorphism φ For a ring A, an automorphism φ of A is said to be inner if there exists an invertible element b of A such that, for any element a of the ring A, the relation φ(a) = bab−1 holds. If A is a ring with automorphism φ, then the following conditions are equivalent. 1) φ is an inner automorphism. 2) There exists an isomorphism π from the ordinary Laurent series ring A((x)) onto the skew Laurent series ring A((y, φ)) which acts on the coefficient ring A identically and preserves lowest degrees of series. ◁ 1) ⇒ 2). Let us have an invertible element b of the ring A such that φ(a) = bab−1 for all a in A. We define a mapping π from the ordinary Laurent series A((x)) ring onto the skew Laurent series ring A((y, φ)) with the use of the rules π(a) = a for all a from A and π(xn ) = b−n yn . Then yn a = π(bn xn a) = π(bn ab−n bn x n ) = φn (a)yn . It is directly verified that the mapping π is a ring homomorphism. The homomorphism π has the inverse ring homomorphism π −1 which is defined by the relations π −1 (a) = a and π −1 (yn ) = bn xn ; therefore, π is an isomorphism. 2) ⇒ 1). Let π be the isomorphism from the assumption. Then the series f = π −1 (y) is of lowest degree 1 and has the form f1 x + f2 x 2 + ⋅ ⋅ ⋅ . The series g = π −1 (y−1 ) is the inverse element of f and is of degree −1. By equating coefficients in the relations fg = 1 and gf = 1, we obtain f1 g−1 = 1 and g−1 f1 = 1. Therefore, g−1 = f1−1 . Then φ(a) = yay−1 = π(f )aπ(g) = π(fag). Since the constant term of the series fag is equal to f1 af1−1 , we obtain the relation φ(a) = f1 af1−1 , as required. ▷ Lemma 1.4. Let A be a ring with automorphism φ and R = A((x, φ)). 1. If M is a maximal right ideal of the ring A, then M((x, φ)) is a maximal right ideal of the ring R. i 2. J(R) ⊆ (J(A))((x, φ)) (i. e., for any series f = ∑∞ i=m fi x ∈ J(R), fi ∈ A, all canonical coefficients fi are contained in J(A)). 3. If J(R) ≠ 0 and B is a non-zero ideal of the ring A generated by the lowest coefficients of the series from J(R), then, for any non-zero element b ∈ B, there exist non-zero elements b󸀠 , b󸀠󸀠 ∈ B such that bb󸀠 = b󸀠󸀠 b = 0. ◁ 1.

It is sufficient to prove that, for every series t ∈ R \ M((x, φ)), there exist two series h ∈ M((x, φ)) and g ∈ R with h + tg = 1. Without loss of generality, we can i assume that t = ∑∞ i=0 ti x , where 0 ≠ t0 ∈ A \ M and ti ∈ A for all i. Since M is a

1.3 The ring A((x, φ)) with inner automorphism φ

| 5

maximal right ideal of the ring A, there exist elements m0 ∈ M and a0 ∈ A such that m0 + t0 a0 = 1. Then ∞

m0 + ta0 = 1 + ∑ ti φi (a0 )x i . i=1

Therefore, there exists a series f ∈ R such that (m0 + ta0 )f = 1. We set h ≡ m0 f ∈ M and g ≡ a0 f ∈ R. Then h + tg = 1. 2. Let {Mi }i∈I be the set of all maximal of right ideals of the ring A. By 1, Mi ((x, φ)) is a maximal right ideal of the ring R for any i. Therefore, J(R) ⊆ ⋂(Mi ((x, φ))) = (J(A))((x, φ)). i∈I

3.

Let f be a non-zero series in J(R) such that b is the lowest coefficient of the series f . We prove that there exists an element b󸀠 ∈ B with bb󸀠 = 0. Since fxn ∈ J(R) for all n, we can assume that the lowest degree of the series f is equal to −1 (i. e., f − bx−1 ∈ A[[x, φ]]). Since f ∈ J(R), the series 1 − f is invertible. Therefore, there exists an i invertible series g ∈ R such that (1 − f )g = 1 and g = 1 + fg. Let g = ∑∞ i=k gi x ∈ R, where gk ≠ 0 and gi ∈ A for all i. Since fg ∈ J(R), all canonical coefficients of the series fg are contained in J(A) by Lemma 1.4(2). Therefore, it follows from the relation g = 1 + fg that the coefficient g0 is invertible in the ring A and bg0 ≠ 0. Since g0 ≠ 0, we have k ≤ 0. Therefore, either k = 0 or k < 0. We assume that k = 0. Then non-zero element bg0 is the lowest coefficient of the series fg and 1 + fg = g. Therefore, the lowest degree k of the series g is equal to −1. This is a contradiction. We assume that k < 0. Since g = 1 + fg, we have bgk = 0 and k is the lowest degree of the series fg ∈ J(R). Therefore, gk ∈ B and we can set b󸀠 = gk . Similarly, we see that there exists an element b󸀠󸀠 ∈ B with b󸀠󸀠 b = 0. ▷

Theorem 1.5 ([59]). If A is a semiprime right Goldie ring, then, for any automorphism φ of the ring A, the skew Laurent series ring A((x, φ)) is semiprimitive. ◁ First, we prove the following assertion (∗). If B is a non-zero ideal of the ring A, then there exists a non-zero element b ∈ B such that bb󸀠 ≠ 0 for any non-zero element b󸀠 ∈ B. By the Zorn lemma, there exists a right ideal C of the ring A such that B∩C = 0 and B ⊕ C is an essential right ideal of the ring A. Since A is a semiprime right Goldie ring, every essential right ideal of the ring A contains a non-zero-divisor [9, 9.13]. Therefore, the essential right ideal B ⊕ C contains some non-zero-divisor b + c, where b ∈ B and c ∈ C. Let b󸀠 be the element of the ideal B with bb󸀠 = 0. Since B is an ideal, cb󸀠 ∈ B ⋂ C = 0. Then (b + c)b󸀠 = bb󸀠 + cb󸀠 = 0. Since b + c is not a zero-divisor, we have b󸀠 = 0. Therefore, b is the required the element. Now Theorem 1.5 follows from Lemma 1.4(3). ▷

6 | 1 Preliminary properties of A((x, φ)) and M((x, φ)) Lemma 1.6. Let A be a ring with automorphism φ, R = A((x, φ)), N an ideal of the ring A with φ(N) = N and φ an automorphism of the factor ring A/N induced by the automorphism φ. 1. If B, C are two arbitrary right ideals of the ring A, then (B+C)((x, φ)) = B((x, φ))+C((x, φ))

and

B((x, φ)) ⋂ C((x, φ)) = (B ⋂ C)((x, φ)).

Consequently, the lattice of right ideals of the ring A is isomorphic to a sublattice of the right ideal lattice of the ring R. 2. There exists a natural ring isomorphism R/(N((x, φ))) ≅ (A/N)((x, φ)). 3. If the ideal N of the ring A is nilpotent, then N((x, φ)) is a nilpotent ideal of the ring R (in particular, N((x, φ)) ⊆ J(R)). 4. If the ring (A/N)((x, φ)) is semiprimitive, then J(R) ⊆ N((x, φ)). 5. If the ring (A/N)((x, φ)) is semiprimitive and N((x, φ)) ⊆ J(R), then J(R) = N((x, φ)). 6. If R is right finite-dimensional, then A is right finite-dimensional. 7. If the rings R and (A/N)((x, φ)) are right finite-dimensional, then the rings A and A/N are right finite-dimensional. ◁ 1, 2, 3. The assertions are directly verified. 4. By 2, if (A/N)((x, φ)) is a semiprimitive ring, then R/(N((x, φ))) is a semiprimitive ring. Therefore, N((x, φ)) ⊇ J(R). 5. The assertion follows from 4. 6. The assertion follows from 1. 7. The assertion follows from 6 and 2. ▷ Remark 1.7. We need the following well-known fact; for example, see [9, Corollary 9.13]. For a ring A, the following conditions are equivalent. 1) A is a semiprime right Goldie ring. 2) A is a right nonsingular, right finite-dimensional, semiprime ring. 3) In the ring A, the set of all essential right ideals coincides with set of all right ideals containing non-zero-divisor. Lemma 1.8. Let A be a ring with automorphism φ, R = A((x, φ)) and N the prime radical of the ring A. 1. φ(N) = N. 2. If A/N is a right finite-dimensional right nonsingular ring and N((x, φ)) ⊆ J(R), then J(R) = N((x, φ)). 3. If the prime radical N is nilpotent and A/N is a right finite-dimensional right nonsingular ring, then the Jacobson radical J(R) of the ring A is nilpotent and J(R) = N((x, φ)).

1.9 The group [[x −1 , x]]A[[x, x −1 ]] and some its subgroups |

7

4. Let the prime radical N be nilpotent, A a ring with maximum condition on right annihilators, and A/N a right finite-dimensional ring. Then the Jacobson radical J(R) of the ring R is nilpotent and J(R) = N((x, φ)). ◁ 1.

For any prime ideal P of the ring A, it is true that the ideals φ(P) and φ−1 (P) are prime and P = φ(φ−1 (P)). In addition, N is the intersection of all prime ideals of the ring A. Therefore, φ(N) = N. 2. Let φ be an automorphism of the factor ring A/N induced by the automorphism φ. Since A/N is a right finite-dimensional, right nonsingular semiprime ring, it follows from Theorem 1.5 and Remark 1.7 that the ring (A/N)((x, φ)) is semiprimitive. By Lemma 1.6(5), J(R) = N((x, φ)). 3. By Lemma 1.6(3), N((x, φ)) is a nilpotent ideal of the ring R. In particular, N((x, φ)) ⊆ J(R). By 2, J(R) = N((x, φ)). 4. The assertion follows from 3 and the following fact: if A is an arbitrary ring with maximum condition on right annihilators, then its factor ring modulo the prime radical is right nonsingular [18]. ▷

1.9 The group [[x −1 , x]]A[[x, x −1 ]] and some its subgroups Let A be an additive Abelian group and x a formal variable. We denote by [[x−1 , x]]A[[x, x−1 ]] the additive Abelian group consisting of all formal series ∑m,n∈ℤ xm amn xn , amn ∈ A, with addition, ∑ xm amn xn + ∑ xm a󸀠mn xn = ∑ x m (amn + a󸀠mn )xn .

m,n∈ℤ

m,n∈ℤ

m,n∈ℤ

Instead of x0 axn , xm ax 0 and x0 ax 0 , we will write axn , x m a and a, respectively. We cannot distinguish the zero elements 0 and ∑m,n∈ℤ x m 0x n of the groups A and [[x−1 , x]]A[[x, x−1 ]]. We denote by A[[x, x−1 ]] and [[x−1 , x]]A the subgroups in [[x −1 , x]]A[[x, x −1 ]] formed by series ∑n∈ℤ an xn and ∑m∈ℤ xm am , respectively. We denote by A((x)) the subgroup in A[[x, x −1 ]] consisting of all series f (x) = ∑n∈ℤ an xn such that an = 0 for almost all negative n. The elements f (x) of the group A((x)) are called left Laurent series in the variable x over A with coefficients an ∈ A. The least subscript n with an ≠ 0 is called the lowest degree of the series f (x); the corresponding monomial an x n and coefficient an are called the lowest term and the lowest coefficient of the series f (x), respectively. The lowest coefficient is denoted by λ(f ) and we assume λ(0) = 0 by definition. For every subset F ⊆ A((x)), we denote by λ(F) the subset {λ(f ) | f ∈ F} ⊆ A. We denote by A[[x]] the subgroup in A((x)) ⊂ A[[x, x−1 ]] consisting of all series ∑n∈ℤ an xn with n ≥ 0. Elements of the group A[[x]] are called left power series in x over A.

8 | 1 Preliminary properties of A((x, φ)) and M((x, φ)) We denote by A((x−1 )) the subgroup in A[[x, x −1 ]] consisting of all series f (x) = ∑n∈ℤ an xn such that an = 0 for almost all positive integers n and the largest subscript n with an ≠ 0 is called the leading degree of the series f (x). The corresponding monomial an xn and coefficient an are called the leading term and the leading coefficient of the series f (x), respectively; the coefficient a0 is called the constant term of the series f (x). We denote by A[[x−1 ]] the subgroup in A((x−1 )) ⊂ A[[x, x −1 ]] consisting of all series ∑n∈ℤ an xn such that n ≤ 0. We denote by A[x, x−1 ] the subgroup A((x)) ∩ A((x−1 )) in A((x)) and A((x−1 )). The elements of the group A[x, x−1 ] are called left Laurent polynomials in x over A; they consist of all f (x) = ∑n∈ℤ an xn such that an = 0 for almost all n. We denote by A[x] the subgroup in A[x, x −1 ] consisting of all f (x) = ∑n∈ℤ an x n such that n ≥ 0 and an = 0 for almost all n. The elements of the group A[x] are called left polynomials in x over A. The groups ((x))A, [[x]]A ((x−1 ))A, [[x −1 ]]A, [x −1 , x]A and [x]A are defined similar to the groups A((x)), A[[x]], A((x−1 )), A[[x−1 ]], A[x, x −1 ] and A[x], respectively. The elements of the groups ((x))A, [[x]]A, [x −1 , x]A and [x]A are called right Laurent series, right power series, right Laurent polynomials and right polynomials in x over A, respectively; similar to the left-side case, we define the lowest degree, the leading degree, the lowest term, the leading term, the lowest coefficient, the leading coefficient and the constant term. We will also sometimes write Ar ((x)), Ar [[x]], Ar [x, x −1 ], Ar [x] instead of ((x))A, [[x]]A, [x−1 , x]A and [x]A, respectively. We denote by [x−1 ]A[[x]] the subgroup in [[x −1 , x]]A[[x, x −1 ]] consisting of all series ∑m,n∈ℤ xm amn xn such that m ≤ 0, n ≥ 0 and amn = 0 for almost all m. We denote by [[x−1 , x]]A[x−1 ] the subgroup in [[x−1 , x]]A[[x, x−1 ]] consisting of all series ∑m,n∈ℤ xm amn xn such that n ≤ 0 and amn = 0 for almost all n.

1.10 Rings A[[x, φ]] and S −1 A[[x, φ]], [[φ, x]]A Let A be a ring with injective endomorphism φ, x a formal variable, and S = {xk }∞ k=0 . We denote by A[[x, φ]] or Aℓ [[x, φ]] the left skew power series ring consisting of n formal series ∑∞ n=0 an x in the variable x with canonical coefficients an ∈ A, where addition is defined naturally and multiplication is defined with the use of the rule xn a = φn (a)xn , a ∈ A, n = 0, 1, 2, . . . . We denote by [[φ, x]]A or Ar [[x, φ]] the right skew power series ring consisting of n formal series ∑∞ n=0 x an in the variable x with canonical coefficients an ∈ A, where addition is defined naturally and multiplication is defined with the use of the rule axn = xn φn (a), a ∈ A, n = 0, 1, 2, . . . . The rings A[[x, φ]] and [[φ, A]] contain as unitary subrings the left skew polynomial ring A[x, φ] and the right skew polynomial ring [φ, x]A consisting of series with a finite number of non-zero coefficients.

1.10 Rings A[[x, φ]] and S−1 A[[x, φ]], [[φ, x]]A

| 9

It is directly verified that S is a left Öre subset of the left skew series ring A[[x, φ]] and S is a right Öre subset of the right skew series ring [[φ, x]]A. A generalized left skew Laurent series ring is a left ring of fractions S−1 A[[x, φ]] of the ring A[[x, φ]] with respect to the left Öre set S. A ring S−1 A[[x, φ]] consists of formal expressions of the form ∞

n

i=0

i=0

x−k ( ∑ ai xi ) = ∑ (x −k ai xi ),

k ≥ 0,

ai ∈ A.

The ring S−1 A[[x, φ]] contains as unitary subring the left ring of fractions S−1 A[x, φ] of the left skew polynomial ring A[x, φ] with respect to the left Öre set S; the ring S−1 A[x, φ] is called the generalized left skew Laurent polynomial ring. A generalized right skew Laurent series ring [[φ, x −1 , x]]AS−1 and a generalized right skew Laurent polynomial ring [φ, x−1 , x]AS−1 are similarly defined. If φ is an automorphism of the ring A, then the generalized left skew Laurent series ring S−1 A[[x, φ]] coincides with the left skew Laurent series ring A((x, φ)) and the generalized left ring skew Laurent polynomials S−1 A[x, φ] coincides with the left skew Laurent polynomial ring A(x, φ). Similar assertions are true for right skew Laurent series rings ((φ, x))A = [[φ, x]]AS−1 and right skew Laurent polynomial rings (φ, x)A = [φ, x]AS−1 . If φ is an automorphism of the ring A, then there are ring isomorphisms ((φ, x))A → A((x, φ−1 )), [[φ, x]]A → A[[x, φ−1 ]], (φ, x)A → A(x, φ−1 ), [φ, x]A → A[x, φ−1 ],

∑ xi ai → ∑ φi (ai )x n ,

∑ xi ai → ∑ φi (ai )x n ,

∑ xi ai → ∑ φi (ai )x n ,

∑ xi ai → ∑ φi (ai )x n ,

which for φ = 1A turn into natural ring isomorphisms ((x))A → A((x)), [[x]]A → A[[x]], (x)A → A(x) and [x]A → A[x] under which the sums ∑ x i ai pass to the sums ∑ ai x i . For φ = 1A , we have A((x, 1A )) = ((1A , x))A = A((x)) = ((x))A, A[[x, 1A ]] = [[1A , x]]A = A[[x]] = [[x]]A, A(x, 1A ) = (1A , x)A = A(x) = (x)A, A[x, 1A ] = [1A , x]A = A[x] = [x]A.

2 Noetherian rings A((x, φ)) Proposition 2.1. Let A be a ring with maximum condition on right annihilators, φ its automorphism, and N the prime radical of the ring A. 1. If A and A/N are right finite-dimensional rings, then the Jacobson radical of the ring A((x, φ)) is nilpotent and coincides with N((x, φ)). 2. If all factor rings of the ring A((x, φ)) are right finite-dimensional, then the Jacobson radical of the ring A((x, φ)) is nilpotent and coincides with N((x, φ)). ◁ 1.

Since A is a right finite-dimensional ring with maximum condition on right annihilators, the nil-ideal N is nilpotent [22]. By Lemma 1.8(4), the Jacobson radical of the ring A((x, φ)) is nilpotent and coincides with N((x, φ)). 2. It follows from Lemma 1.6(7) and Lemma 1.6(2) that the rings A and A/N are right finite-dimensional. By 1, the Jacobson radical of the ring A((x, φ)) is nilpotent and coincides with N((x, φ)). ▷ Proposition 2.2. If φ is an automorphism of the ring A, then the following conditions are equivalent. 1) A((x, φ)) is a right Artinian ring. 2) A is a ring with maximum condition on right annihilators and A((x, φ)) is a semilocal ring such that all factor rings of A are right finite-dimensional. ◁ We set R = A((x, φ)). 1) ⇒ 2). It is clear that R is a semilocal ring such that all factor rings are right finitedimensional. Since R is a right Noetherian ring, R is a ring with maximum condition on right annihilators. In addition, any subring of an arbitrary ring with maximum condition on right annihilators has this property. 2) ⇒ 1). By Proposition 2.1(2), J(R) is a nilpotent ideal of the ring R. It follows from the assumption that R/J(R) is a semisimple ring and for every positive integer n, J n (R)/J n+1 (R) is a semisimple Artinian right R-module. Therefore, R is a right Artinian ring. ▷ Proposition 2.3. Let A be a ring with maximum condition on right annihilators, N the prime radical of the ring A, and φ an automorphism of the ring A. 1. If the rings A and A/N are right finite-dimensional, then the Jacobson radical of the ring A((x, φ)) is nilpotent and coincides with N((x, φ)). 2. A((x, φ)) is a right Artinian ring if and only if A((x, φ)) is a semilocal ring and all its factor rings are right finite-dimensional. Proposition 2.3 follows from Proposition 2.1(1) and Proposition 2.2. https://doi.org/10.1515/9783110702248-002

2 Noetherian rings A((x, φ))

| 11

In connection with Proposition 2.3(2), we note that a semilocal ring such that all its factor rings are right finite-dimensional is not necessarily right Artinian. For example, the formal power series ring F[[x]] over any field F is such a ring; we note that the ring F[[x]] is Noetherian. Theorem 2.4 ([59]). If A is a right Noetherian ring with automorphism φ, then the Jacobson radical of the ring A((x, φ)) is nilpotent and coincides with N((x, φ)), where N is the prime radical of the ring A. Theorem 2.4 follows from Proposition 2.3(1). Lemma 2.5. Let A be a ring with automorphism φ, M a right A-module, and M((x, φ)) the corresponding Laurent series A((x, φ))-module. 1. If P is an A((x, φ))-submodule in M((x, φ)) and there exist series f1 , f2 , . . . , fn in P such that fi = fi,0 + fi,1 x + fi,2 x 2 + ⋅ ⋅ ⋅

for every i and gm ∈f1,0 A + f2,0 A + ⋅ ⋅ ⋅ + fn,0 A for every series g = gm x m + gm+1 x m+1 + ⋅ ⋅ ⋅ from P, where all fi,k , gk are contained in A, then the A((x, φ))-module P is generated by n series f1 , f2 , . . . , fn . In particular, if M is a Noetherian Bezout A-module, then M((x, φ)) is a Noetherian Bezout A((x, φ))-module. 2. M((x, φ)) is a simple A((x, φ))-module if and only if M is a simple A-module. ◁ We denote R = A((x, φ)) and N = M((x, φ)). 1. We denote by Q the submodule f1 R + f2 R + ⋅ ⋅ ⋅ + fn R in PR . Since all series fi are contained in the module P and Q = f1 R + f2 R + ⋅ ⋅ ⋅ + fn R, we have Q⊆P. We assume that the assertion of Lemma 2.5(1) is not true. Then there exists a series h∈P such that h∈Q. ̸ Without loss of generality, we can assume that h = h0 + h1 x + h2 x2 + ⋅ ⋅ ⋅ . By assumption, h0 = f1,0 a1,0 + f2,0 a2,0 + ⋅ ⋅ ⋅ + fn,0 an,0 for some elements a1,0 , . . . , an,0 of the ring A. We consider the series h − (f1 a1,0 + ⋅ ⋅ ⋅ + fn an,0 ) = h󸀠 = h󸀠1 x + h󸀠2 x 2 + ⋅ ⋅ ⋅ . The series h󸀠 is contained in the module P as well, and the assumption of the lemma is applicable to it: h󸀠1 = f1,0 a1,1 + f2,0 a2,1 + ⋅ ⋅ ⋅ + fn,0 an,1 . We set h󸀠󸀠 = h󸀠 − (f1 a1,1 + ⋅ ⋅ ⋅ + fn an,1 )x. The sequence h, h󸀠 , h󸀠󸀠 , . . . can be continued to infinity and the degree of the lowest term of the elements of the sequence will be increasing. It is directly verified that h = f1 (a1,0 + a1,1 x + a1,2 x2 + ⋅ ⋅ ⋅) + ⋅ ⋅ ⋅ + fn (an,0 + an,1 x + ⋅ ⋅ ⋅). This contradicts the assumption h∈Q. ̸

12 | 2 Noetherian rings A((x, φ)) 2. Let M be a no-simple A-module. Then it contains a proper non-zero submodule L. Therefore, the module NR contains a proper non-zero submodule L. Now let MA be a simple module. It is sufficient to prove that NR = fR for any nonzero series f in N. Without loss of generality, we can assume that f = f0 + f1 x + f2 x2 + ⋅ ⋅ ⋅ , where f0 is non-zero, so M = f0 A. Then f0 A = M = λ(N) and we can apply 1 to the module N and the series system consisting of the unique series f , which completes the proof. ▷ Lemma 2.6. Let A be a ring with automorphism φ and R = A((x, φ)) the skew Laurent series ring. 1. The ring R is semisimple if and only if the ring A is semisimple. 2. The inclusion J(R)⊆J(A) holds. If the radical J(A) of the ring A is nilpotent, then J(R) = J(A) is a nilpotent ideal of the ring R. 3. If I is a two-sided ideal of the ring A and φ(I) = I, then R/I≅(A/I)((x, φI )). 4. If the ring R is semilocal, then the ring A is semilocal. 5. If the ring A right Noetherian and I⊆J are two right ideals of the ring R, then the relation λ(I) = λ(J) implies the relation I = J. 6. If the ring A is right Noetherian, then the ring R is right Noetherian. 7. If A is a reduced ring and φ(I) = I for any right ideal I of the ring A, then J(R) = 0. ◁ 1.

Let the ring R be semisimple. It is sufficient to prove that the ring A is regular and does not contain an infinite set pairwise orthogonal idempotents. Since the ring R is regular, for any element a of the ring A, we can find the series f ∈R with afa = a. Consequently, af0 a = a, where f0 ∈A. Therefore, A is a regular ring. Now we assume that the ring A contains an infinite system of pairwise orthogonal idempotents. Since this system is contained in the ring R, this contradicts the semisimplicity of the ring R. Now let the ring A be semisimple. Then it is the finite direct sum of its minimal right ideals Ii . By Lemma 2.5(2), the right ideals Ii of the ring R are minimal as well. Then the ring R is semisimple, since R = ∑Ii . 2. By Lemma 1.4(1), if I is a maximal right ideal of the ring A, then I is a maximal right ideal of the ring R. This implies that J(R)⊆J(A). The set J(A) is a two-sided ideal of the ring R, since φ(J(A)) = J(A). Now we assume that the ideal J(A) is nilpotent. Then the ideal J(A) is nilpotent as well. Consequently, J(R)⊇J(A), which completes the proof. 3. Indeed, I is a two-sided ideal of the ring R, since φ(I) = I. The remaining part of the assertion is directly verified. 4. Indeed, the ring R/J(R) is semisimple. By 2, R/J(A) is isomorphic to a factor ring of the semisimple ring R/J(R); consequently, it is semisimple as well. Since φ(J(A)) = J(A), we can use 3. We find that the ring (A/J(A))((x, φJ(A) )) is semisimple. By 1, the ring A/J(A) is semisimple as well.

2 Noetherian rings A((x, φ))

| 13

5.

The right ideal λ(I) is finitely generated, since the ring A is Noetherian. Let {a1 , . . . , an } be a generator system of the right ideal λ(I). For every i, we choose a series fi ∈I such that its lowest coefficient is equal to ai . For every i, we can choose a series fi such that the degree of its lowest term is equal to the zero. By applying to the ideal I Lemma 2.5(1), we obtain I = f1 R + f2 R + ⋅ ⋅ ⋅ + fn R. Similarly, J = f1 R + f2 R + ⋅ ⋅ ⋅ + fn R, since all fi are contained in J. 6. Let {Ii } be an ascending chain of right ideals in the ring R. The ascending chain {λ(Ii )} of right ideals of the right Noetherian ring A stabilizes. By 5, the chain {Ii } stabilizes as well. 7. In the proof, we will use the following property of a reduced ring A: if a, b ∈ A and ab = 0, then (ba)2 = b(ab)a = 0. Therefore ba = 0, since the ring A has no nonzero nilpotent elements. In addition, every right invertible element of the reduced ring A is left invertible. Now we assume that the radical J(R) is non-zero. Then it contains a series f such that degree of the lowest term is equal to −1. It follows from 2 that all coefficients of the series f are contained in the radical J(A). Let g be the series in R, which is inverse to 1 + f , and let gm xm be the lowest term of the series g. We assume that m < 0. By equating coefficients in the relation (1 + f )g = 1, we obtain the relations f−1 φ−1 (gm ) = 0 and f−1 φ−1 (gm+1 ) + (1 + f0 )gm = 0. It follows from the first relation that f−1 φ−1 (gm )A = 0. Then f−1 φ−1 (gm A) = 0. Therefore, (with the use of φ−1 (gm A) = gm A) we obtain the relation f−1 gm A = 0. Therefore, f−1 gm = 0. Therefore, gm f−1 = 0. By multiplying the second relation by f−1 , we obtain the relation f−1 φ−1 (gm+1 )f−1 = 0. Then (f−1 φ−1 (gm+1 ))2 = 0. Since the ring A is reduced, we have f−1 φ−1 (gm+1 ) = 0. Therefore, by considering the second relation, we obtain (1 + f0 )gm = 0. In addition, the element 1 + f0 is invertible in the ring A, since f0 ∈ J(A). Then gm = 0, which contradicts the choice of the coefficient gm . Now we assume that m = 0. Similar to the previous case, we obtain f−1 φ−1 (gm ) = 0

and f−1 φ−1 (gm+1 ) + (1 + f0 )gm = 1.

Similar to the previous case, f−1 gm = gm f−1 = 0. It follows from the second relation that (1 + f0 )gm = 1 − f−1 φ−1 (gm+1 ). The right part of this relation is invertible in A, since f−1 is contained in J(A). Therefore, (1 + f0 )gm is an invertible element of the ring A. Then the element gm is invertible in A as well, which contradicts the relation gm f−1 = 0 and the choice of the series f . Finally, we assume that m > 0. Then f−1 φ−1 (g1 ) = 1; this is impossible, since the element f−1 is contained in radical J(A) and cannot be invertible. ▷

14 | 2 Noetherian rings A((x, φ)) Remark 2.7. Let A be a ring, P its right ideal, and a a right invertible element of the ring A. If a is contained in set 1 + P, then a−1 is contained in 1 + P as well. Indeed, a−1 = 1 + (a−1 − 1) = 1 + (a−1 − aa−1 ) = 1 + (1 − a)a−1 ∈ 1 + P. Theorem 2.8 ([68]). Let A be a ring with the Jacobson radical J and Ja ⊆ aA for any element a ∈ J. Then the lowest coefficient of any series f in the Jacobson radical J(A((x))) of the Laurent series ring A((x)) generates a nilpotent right ideal in the ring A. ◁ Let f be an arbitrary series from J(A((x))) with lowest coefficient fk . We denote by f 󸀠 the series f − fk xk . Then the lowest degree of the series f 󸀠 is not less that k + 1. In addition, J(A((x))) ⊆ J by Lemma 2.6(2). Therefore, all coefficients of series f and f 󸀠 are contained in J. Then the series 1 − f 󸀠 x−k−1 is of lowest degree 0. Its lowest coefficient is equal to 1−fk+1 ; consequently, it is invertible. By 1.2(7), the series 1−f 󸀠 x−k−1 is invertible and its inverse series h is contained in 1 + J by Remark 2.7. The lowest coefficient h0 of the series h is contained in 1 + J; consequently, it is invertible. We consider the series (1−fx−k−1 )h. On the one hand, it is the product of two invertible series; consequently, it is invertible. On the other hand, it is equal to (1 − f 󸀠 x −k−1 − fk x−1 )h = 1 − fk hx−1 . It is easy to see that the lowest term of this series is equal to fk h0 x−1 . Let g be the series which is inverse to series 1 − fk hx−1 . It is easy to see that its lowest degree cannot be positive, since otherwise the coefficient of x 0 of the product g(1 − fk hx−1 ) is equal to g1 fk h0 ∈ J, which is impossible, since it has to be equal to 1. Let the lowest degree of the series g be equal to −n + 1, where n is a positive integer. We consider the relation 2

n

(1 + fk hx−1 + (fk hx−1 ) + ⋅ ⋅ ⋅ + (fk hx−1 ) )(1 − fk hx−1 ) = 1 − (fk hx−1 )

n+1

.

By multiplying both of its parts by g from the right, we obtain 2

n

n+1

1 + fk hx−1 + (fk hx−1 ) + ⋅ ⋅ ⋅ + (fk hx−1 ) = g − (fk hx −1 )

g.

By equating coefficients of x−n in the left part and the right part, we find (considering the property that the lowest degree g exceeds −n) that the element (fk h0 )n is contained in the right ideal generated by elements of the form (fk hi1 )(fk hi2 ) ⋅ ⋅ ⋅ (fk hin+1 ), where i1 , i2 , . . . , in+1 are arbitrary non-negative integers. We denote fk h0 by a, a ∈ J, then we find that the element an is contained in the right ideal generated by elements of the form ah0 −1 hi1 ah0 −1 hi2 ⋅ ⋅ ⋅ ah0 −1 hin+1 . In addition, for all non-zero k, we find that hk is contained in J. We prove that all elements ah0 −1 hi1 ah0 −1 hi2 ⋅ ⋅ ⋅ ah0 −1 hin+1

2 Noetherian rings A((x, φ))

| 15

are contained in the right ideal of the ring A generated by the element an+1 . Indeed, if i1 is not equal to 0, then the inclusion hi1 ah0 −1 hi2 ⋅ ⋅ ⋅ ah0 −1 ∈ ah0 −1 hi2 ⋅ ⋅ ⋅ ah0 −1 A holds by the assumption. By applying this condition to all non-zero ik , we obtain the required inclusion ah0 −1 hi1 ah0 −1 hi2 ⋅ ⋅ ⋅ ah0 −1 hin+1 ∈ an+1 A. Therefore, we obtain an ∈ an+1 A, i. e., an = an+1 b, where b is some element of the ring A. Then an (1 − ab) = 0. Since the element a is contained in the Jacobson radical J of the ring A, the element 1 − ab is invertible; consequently, an = 0. Now we prove that aA is a nilpotent right ideal of the ring A. Indeed, since AaA is contained in J, we find that (aA)2n is contained in aJaJa . . . JaA, where a is repeated n times in the expression. By assumption, JaJ is contained in aAJ = aJ and Ja is contained in aA. By applying these inclusions the required number times, we find that (aA)2n is contained in an A = 0, i. e., aA = ah0 −1 A = fk A is a nilpotent right ideal, as required. ▷ Theorem 2.9 ([68]). Let A be a ring such that J(A)a ⊆ aA for all a ∈ J(A). Then the Jacobson radical J(A((x))) of the Laurent series ring A((x)) coincides with ideal N((x)) formed by series whose canonical coefficients are contained in the prime radical N of the ring A. ◁ We denote J = J(A). We recall that, for any subset P of the ring A, we denote by P the set of series from A((x)) whose coefficients are contained in P. In addition, the Jacobson radical of the ring A((x)) is contained in J by Lemma 2.6(2). First, we prove that, if a is an arbitrary element of the prime radical N, then the right ideal aA is nilpotent. Indeed, the element a is nilpotent, let an = 0. It follows from the property that AaA is contained in N ⊆ J that (aA)2n is contained in aJaJa . . . JaA, where a is repeated in the expression n times. By assumption, JaJ is contained in aAJ = aJ and Ja is contained in aA. By applying these inclusions the required number times, we find that (aA)2n is contained in an A = 0, as required. If the right ideal aA is nilpotent, then the right ideal aA((x)) is nilpotent as well. Therefore, an arbitrary element of the prime radical N of the ring A is contained in the prime radical of the ring A((x)) as well; therefore, it is contained in the Jacobson radical of the ring A((x)). Now let f be an arbitrary series from the Jacobson radical of the ring A. By Theorem 2.8, its lowest coefficient fk is contained in N; by the above, it is contained in the Jacobson radical of the ring A((x)). Then the series f 󸀠 = f − fk x k is contained in the Jacobson radical of the ring A. By applying the same argument to the series f 󸀠 , we prove that fk+1 is contained in N. Continuing, we find that all coefficients of the series f are

16 | 2 Noetherian rings A((x, φ)) contained in N. Therefore, it is proved that the Jacobson radical of the ring A((x)) is contained in N. Now we prove the converse inclusion. It is sufficient to prove that, if f is an arbitrary series from N, then the series 1 + f is right invertible. If the lowest degree of the series f is not negative, then this property follows from 1.2(7). Now let the lowest degree of the series f be negative; we denote it by −n. Then we consider the series f 󸀠 = f0 + f1 x + f2 x2 + ⋅ ⋅ ⋅ . By 1.2(7), the series 1 + f 󸀠 is invertible; therefore, it is sufficient to prove that the series (1 + f )(1 + f 󸀠 )−1 = 1 + (f − f 󸀠 )(1 + f 󸀠 )−1 is invertible. However, the series f −f 󸀠 is the finite sum of f−n x−n +⋅ ⋅ ⋅+f−1 x −1 . Each of the elements f−i are contained in N and, as is proved above, in J(A((x))); therefore, the series f − f 󸀠 is contained in J(A((x))) as well. Therefore, we find that the series (1 + f )(1 + f 󸀠 )−1 = 1 + (f − f 󸀠 )(1 + f 󸀠 )−1 is invertible, which we were required to prove. ▷ Corollary 2.10 ([68]). Let A be a ring with prime radical N. 1. If any two elements of the radical J(A) of the ring A commute with respect to multiplication, then the Jacobson radical J(A((x))) of the Laurent series ring A((x)) coincides with the ideal N((x)). 2. If the ring A is right or left invariant, then the Jacobson radical J(A((x))) of the Laurent series ring A((x)) coincides with the ideal N((x)). Corollary 2.10 follows from Theorem 2.9.

3 Serial and Bezout rings A((x, φ)) Remark 3.1. Let R be a right serial ring. Then R is a semilocal ring and any factor ring of the ring R is the finite direct sum of uniserial right ideals. It is well known that an arbitrary ring A is right finite-dimensional if and only if A has the essential right ideal which is the finite direct sum of right uniform ideals. Therefore, every factor ring of the ring R is right finite-dimensional. Proposition 3.2. If A is a ring with automorphism φ, then the following conditions are equivalent. 1) A((x, φ)) is a right Artinian right serial ring; 2) A is a right Artinian ring right serial ring; 3) A((x, φ)) is a right serial ring with maximum condition on right annihilators; 4) A((x, φ)) is a right serial ring and A is a ring with maximum condition on right annihilators; 5) A((x, φ)) is a right serial ring and A is a semiprimary ring. ◁ We denote R = A((x, φ)) and J = J(R). 1) ⇒ 2). The mapping I → I is an injective lattice homomorphism from the lattice of right ideals of the ring A in the lattice of right ideals of the right Artinian ring R. Therefore, the ring A is right Artinian. Therefore, radical J is nilpotent. By Lemma 2.6(2), J(R) = J. We consider the semisimple ring B = A/J and some its system of primitive pairwise orthogonal idempotents {ei } with ∑ ei = 1. The automorphism φ of the ring naturally induces an automorphism φJ of the ring B. All modules ei B are simple. By Lemma 2.5(2), the corresponding right ideals ei B((x, φJ )) of the ring B((x, φJ )) also are simple of the modules. Therefore, the idempotents ei are primitive in the semisimple ring B((x, φJ )) as well. By Lemma 2.6(3), the ring B((x, φJ )) is isomorphic to the ring R/J(R). Since the ideal J is nilpotent, the finite system of pairwise orthogonal idempotents {ei } in the ring A/J can be lifted to a system of primitive pairwise orthogonal idempotents {di } of the ring A with ∑ di = 1; see, for example, [57, 4.19(1)]. Then all idempotents di are also primitive in the ring R, since their images in the ring R/J(R) are primitive. Since R is a right serial ring and any direct decomposition of the right Artinian ring R into a direct sum of non-zero indecomposable right ideals is unique up to isomorphism, all right ideals di R of the ring R are uniserial of the modules over R. In addition, for every i, the submodule lattice of the right ideal di A is embedded in the submodule lattice of di R with the use of the mapping M → M. Therefore, the module di A is uniserial as well, and the ring A can be represented in the form of the direct sum of a finite number of uniserial modules di A. Consequently, A is a right serial ring. ▷ 2) ⇒ 1). Since A is a right serial ring, it contains a finite system of pairwise orthogonal idempotents ei , i = 1, . . . , n, such that ∑ ei = 1 and ei A is a uniserial module over https://doi.org/10.1515/9783110702248-003

18 | 3 Serial and Bezout rings A((x, φ)) A for every i. For every idempotent e and any two-sided ideal I the relation eA ∩ I = eI holds. In particular, for all i = 1, . . . , n and any non-negative integer k, the relation ei A ∩ J k = ei J k holds (we assume that J 0 = A). Since A is a right Artinian ring, the ideal J is nilpotent and the ring A/J is semisimple. Consequently, all modules J k /J k+1 over A/J are semisimple as well. Therefore, for all i and k, the uniserial module ei J k /ei J k+1 ≅ (ei J k + J k+1 )/J k+1 ⊆ J k /J k+1 is a simple A/J-module; consequently, it is a simple A-module. For every i, the module ei A is uniserial. Therefore, the modules ei J k are only its submodules and only finite number of these modules are non-zero, since J is a nilpotent ideal. In addition, every module ei J k is generated by any its element not contained in the module ei J k+1 . Now we consider the Jacobson radical J(R) of the ring R. The radical J of the ring k

right Artinian A is nilpotent and J(R) = J by Lemma 2.6(2). It is obvious that J k = J for any positive integer k. Now we consider right ideals ei R = ei A of the ring R. The direct sum of them is equal to R. Now we prove that, for every i, the right ideal ei R is a uniserial module over R. Let f ∈ ei R be a non-zero series in R. For some non-negative integer k, the inclusion f ∈ ei J k \ ei J k+1 holds. Since the module ei J k /ei J k+1 is simple, the right ideal fR contains a series g such that all its coefficients, except g0 , are contained in ei J k+1 and g0 is contained in set ei J k \ ei J k+1 . Then the element g0 generates the whole right ideal ei J k . Therefore, g = g0 (1 + h), where h ∈ J = J(R). Therefore, fR = gR = g0 R = ei J k . Therefore, the modules ei J k are only submodules of the module ei R and ei R is a uniserial Artinian module. Consequently, the module RR is the finite direct sum of uniserial Artinian modules. Therefore, R is a right serial right Artinian ring, as required. The implications 1) ⇒ 3) and 2) ⇒ 5) are obvious. The implication 3) ⇒ 4) follows from the property that every subring of the ring with maximum condition on right annihilators is a ring with maximum condition on right annihilators. The implication 4) ⇒ 3) follows from Proposition 2.2 and Remark 3.1. 5) ⇒ 4). Let J = J(A). By assumption, A/J is a semisimple ring and the ideal J is nilpotent. We take any positive integer n. It is directly verified that J n−1 /J n is a semisimple right A-module, φ(J n ) = J n , an automorphism φ induces automorphism φn factor rings A/J n and there exists a natural ring isomorphism (A/Jn )((x, φn )) ≅ A((x, φ))/Jn ((x, φ)). Since the ring (A/Jn )((x, φn )) is isomorphic to a factor ring of the right serial ring A((x, φ)), we find that (A/Jn )((x, φn )) is a right serial ring. Since the ring A((x, φ)) is right finite-dimensional, the ring A is right finite-dimensional by Lemma 1.6(6). For any positive integer n, the finite-dimensional semisimple right A-module J n−1 /J n is an Artinian module. In addition, the ideal J is nilpotent. Therefore, A is a right Artinian ring. In particular, A is a right Noetherian ring. ▷

3 Serial and Bezout rings A((x, φ))

| 19

Corollary 3.3. If A is a ring with automorphism φ and A((x, φ)) is a right serial ring, then A/Sing AA is a right Artinian right serial ring.

◁ We set S = Sing AA and A = A/S. Since the ring A((x, φ)) is right finite-dimensional,

the ring A is right finite-dimensional by Lemma 1.6(6). Therefore, A is a ring with max-

imum condition on right annihilators, [50]. For any essential right ideal B of the ring A, the right ideal φ(B) is essential as well. Therefore, φS = S and φ induces the auto-

morphism φ of the ring A. Since there exists a natural ring isomorphism A((x, φ)) ≅

A((x, φ))/S((x, φ)), the ring A((x, φ)) is isomorphic to a factor ring of the right serial

ring A((x, φ)). Therefore, A((x, φ)) is a right serial ring. By Proposition 3.2, A is a right

Artinian right serial ring. ▷

Corollary 3.4. Let A be a ring with automorphism φ and S = Sing AA . 1)

If A((x, φ)) is a right serial ring, then the following conditions are equivalent. The ideal S is nilpotent.

2) S is an Artinian right A-module.

3) S is a right Noetherian A-module.

4) A is a right Artinian right serial ring. ◁ By Corollary 3.3, A/S is a right Artinian right serial ring. In particular, A/S is a right Noetherian ring.

1) ⇒ 2). Since the ideal S is nilpotent and A/S is a right Artinian ring, A is a semipri-

mary ring. By Proposition 3.2, A is a right Artinian ring.

2) ⇒ 3). Since right A-modules A/S and S are Artinian, A is a right Artinian ring. In

particular, A is a right Noetherian ring and SA is a Noetherian module.

3) ⇒ 4). Since right A-modules A/S and S are Noetherian, A is a right Noetherian

ring. By Proposition 3.2, A is a right Artinian ring right serial ring. The implication 4) ⇒ 1) is directly verified. ▷

Theorem 3.5 ([60]). If A is a ring with automorphism φ, then the following conditions are equivalent. 1)

A((x, φ)) is a right serial ring with maximum condition on right annihilators.

2) A((x, φ)) is a right serial right Artinian ring.

3) A((x, φ)) is a right serial ring and A is a ring with maximum condition on right annihilators.

4) A((x, φ)) is a right serial ring and the right singular ideal of the ring A is nilpotent.

5) A is a right serial right Artinian ring. Theorem 3.5 follows from Proposition 3.2 and Corollary 3.4.

20 | 3 Serial and Bezout rings A((x, φ))

3.6 Reductable and nonreductable sums of submodules A sum of submodules is said to be reductable if it has a summand such that, when removed from the sum, the sum does not change. A sum of submodules (or ideals) is said to be nonreductable if, when removing any of its terms, the sum changes. 1. If A is a ring and M is a right A-module, then the following conditions are equivalent. i) Any infinite sum of submodules {Mα |α ∈ Ω} of the module MA is reductable. ii) All factor modules of the module MA are finite-dimensional. 2. Let A be a ring with automorphism φ, M a right A-module and A((x, φ)) a skew Laurent series ring. If all factor modules of the right A((x, φ))-module M((x, φ)) are finite-dimensional, then this module is Noetherian. ◁ 1. The idea of the proof is taken from [55]. i) ⇒ ii). Indeed, let N be a factor module of the module M which contains an infinite direct sum of non-zero submodules Nα . Then we consider of their pre-images Mα under the canonical homomorphism from MA onto NA . By assumption, the sums of submodules Mα have to be reductable, i. e., for some β, the module Mβ is contained in the sum ∑α=β̸ Mα . However, then the module Nβ is contained in the sum ∑α=β̸ Nα , which contradicts the assumption. ii) ⇒ i). Let {Mα |α ∈ Ω} be an infinite set of submodules of the module M. We consider the submodule P of the module M which is the sum ∑ (Mβ ⋂ ∑ Mα ).

β∈Ω

α=β ̸

Let p = ∑β∈Ω mβ , where only a finite number of terms in the sum is non-zero, be the element of the submodule P such that mβ ∈ Mβ ⋂ ∑α=β̸ Mα for all β in Ω. For every β ∈ Ω, the element mβ is contained in the module ∑α=β̸ Mα and the element p − mβ = ∑α=β̸ mα is contained in the module ∑α=β̸ Mα . Therefore, the element p is contained in the module ∑α=β̸ Mα as well. Therefore, the inclusion P ⊆ ∑α=β̸ Mα holds, for every β ∈ Ω. We denote by N the factor module M/P and we denote by Nα the image of the submodule Mα under the canonical homomorphism M onto N. We prove that the modules Nα form a direct sum. We have to prove that, for every β ∈ Ω, the intersection the module Nβ with the module ∑α=β̸ Nα is equal to zero. It is sufficient to prove that the intersection the module Mβ +P with the module P +∑α=β̸ Mα is contained in P. It was proved above that the inclusion P ⊆ ∑α=β̸ Mα holds; therefore, we have to prove only that the intersection of the module Mβ with the module ∑α=β̸ Mα is contained in P. However, this is true by the definition of the module P.

3.6 Reductable and nonreductable sums of submodules | 21

Therefore, the modules Nα form an infinite direct sum in the factor module N = M/P; each of them cannot be nonzero modules by the assumption. Therefore, for some β ∈ Ω, the module Nβ is equal to zero; consequently, Mβ ⊆ P. It was proved above that the inclusion P ⊆ ∑α=β̸ Mα holds; therefore, Mβ ⊆ ∑α=β̸ Mα . However, this means that sum of ∑β∈Ω Mβ is reductable. 2. We set R = A((x, φ)). By 1, it is sufficient to prove that the right R-module M((x, φ)) does not contain infinite nonreductable sums of submodules. We assume that module MA is not Noetherian. Then there exists a strictly ascending chain of its submodules m1 A ⊂ m1 A + m2 A ⊂ m1 A + m2 A + m3 A ⊂ ⋅ ⋅ ⋅ . We construct a system of series f i ∈ M((x, φ)) such that the R-modules f i R are summands of an infinite nonreductable submodule sum in the R-module M((x, φ)). Let g: ℕ → ℕ be a surjective mapping from the set of positive integers onto the set of positive integers and g −1 (n) an infinite set for every n. (We can set, for example, g(n) = n − [√n]2 + 1, where [x] denotes the integral part of x.) We define the series f i ∈ M((x, φ)) as follows: f2g(n) = mn for every n and all ren maining coefficients of all series f i are equal to zero. Now we assume that f i R does not form a nonreductable sum. Then there exists an integer i such that f i ∈ ∑j=i̸ f j R. Therefore, f i = ∑j=i̸ f j rj and this sum contains only a finite number of non-zero terms. We denote by λ the least from degrees of the lowest coefficients rj . By the choice of the mapping g, the set g −1 (i) is infinite. Therefore, this set contains a positive integer n such that 2n > −λ. Then f2in = f2g(n) = mn . It follows from the relation f i = ∑j=i̸ f j rj that mn ∈ n j

j

can be is non-zero only if k coincides ∑j=i,k≤2 n −λ f A. In addition, the coefficient f ̸ k k n

n+1

with the degree of 2 and k ≤ 2 − λ < 2

j

. For k = 2n , the coefficient fk (for j not equal j

to i) is equal to zero as well. Therefore, mn ∈ ∑j=i,k n, where n is the positive integer from 1, then the finitely generated module N is not cyclic. 5. For any Bezout right A-module M, all factor modules of the module M are finitedimensional. 6. If A is a right Bezout ring, then all cyclic right A-modules are finite-dimensional. ◁ 1, 2, 3. These assertions are directly verified. 4. The assertion follows from 3. 5. Since all factor modules of the Bezout module M are Bezout modules, it is sufficient to prove that M is a finite-dimensional module. We assume the contrary. Then the module M contains a submodule X = ⨁∞ i=1 Xi , where all Xi are non-zero cyclic modules. Every non-zero cyclic module Xi has a simple factor module Xi /Yi = Si . We denote by h the natural epimorphism from the Bezout module M onto the factor module M/(⨁∞ i=1 Yi ). The module h(M) contains the direct sum of an infinite number of simple modules h(Xi ). By 4, h(M) contains a finitely generated semisimple submodule which is not cyclic. Therefore, the module h(M) is not a Bezout module. This contradicts the property that every homomorphic image of the Bezout module is a Bezout module. 6. Since all cyclic right A-modules are homomorphic images of the module AA , the assertion follows from 5. ▷

3.9 Distributive modules and rings A module M is said to be distributive if (X + Y) ∩ Z = X ∩ Z + Y ∩ Z for any three its submodules X, Y, Z, i. e., the submodule lattice of the module M is distributive.

3.9 Distributive modules and rings | 23

A ring A is said to be right (resp., left) distributive if the lattice of its right (resp., left) ideals is distributive, i. e., A is a distributive right (resp., left) A-module. Assertions 1 and 2 given below are well known [56]; also see, for example, [57, Assertions 1.17 and 1.19]). We also need the familiar assertion 3; for example, see [57, Theorem 3.22]. For convenience, we give the proof of these assertions. 1. If A is a ring and M is a right A-module, then the module M is distributive if and only if for any two elements x, y of the module M, there exist two elements a, b of the ring A such that a + b = 1 and xaA + ybA ⊆ xA ∩ yA. 2. Let A be a ring, M a distributive right A-module, and X, Y two submodules in M with X ∩ Y = 0. Then, for any two elements x ∈ X and y ∈ Y, there exist two elements a, b of the ring A such that a + b = 1 and xaA = ybA = 0. In addition, there are no non-zero homomorphisms between the modules X and Y. 3. If A is a semilocal ring and M is a distributive right A-module, then M is a quotient finite-dimensional Bezout module. In particular, every right distributive semilocal ring is a right Bezout ring, over which all cyclic right modules are finite-dimensional. 4. If A is a semilocal ring and M is a finite direct sum of distributive right A-modules, then M is a quotient finite-dimensional module. In particular, every cyclic right module over a right semidistributive semilocal ring is a factor-finite-dimensional module. ◁ 1.

Let the module M be distributive and x, y ∈ M. Since (x + y)A = (x + y)A ∩ xA + (x + y)A ∩ yA, there exist two elements b, d ∈ A such that (x + y)b ∈ xA,

(x + y)d ∈ yA,

x + y = (x + y)b + (x + y)d.

Therefore, yb = (x + y)b − xb ∈ xA ∩ yA and xd = (x + y)d − yd ∈ xA ∩ yA. We denote a = 1 − b and z = a − d = 1 − b − d. Then 1 = a + b,

(x + y)z = (x + y) − (x + y)b − (x + y)d = 0,

xa = xd + xz = xd + (x + y)z − yz = xd − yz, yz = −xz ∈ xA ∩ yA,

xa ∈ xA ∩ yA.

Conversely, let X, Y, Z ∈ M and z = x + y ∈ (X + Y) ∩ Z, where x ∈ X and y ∈ Y. By assumption, there exist two elements a, b ∈ A such that 1 = a + b, xa ∈ yA and yb ∈ xA. Then zb = xb + yb ∈ xA ∩ zA, za = xa + ya ∈ yA ∩ zA,

z = zb + za ∈ X ∩ Z + Y ∩ Z,

(X + Y) ∩ Z ⊆ X ∩ Z + Y ∩ Z ⊆ (X + Y) ∩ Z. 2. Since X ∩ Y = 0, we have xA ∩ yA = 0. By 1, there exist two elements a, b ∈ A such that a + b = 1 and xaA, ybA ⊆ xA ∩ yA = 0.

24 | 3 Serial and Bezout rings A((x, φ)) Now let f : X → Y be a module homomorphism, x󸀠 ∈ X, y󸀠 = f (x 󸀠 ) ∈ Y. According to the property proved, there exist two elements a, b ∈ A such that a + b = 1, x 󸀠 a = 0, y󸀠 b = 0. Then f (x󸀠 ) = y󸀠 = y󸀠 a + y󸀠 b = f (x󸀠 a) + 0 = f (0) = 0,

f = 0.

3.

By 3.8(5), it is sufficient to prove that M is a Bezout module. Let N be a finitely generated submodule of the module M and J = J(A). Then N/NJ is the finite direct sum of simple modules S1 , . . . , Sk . Since N/NJ is a distributive module, all modules Si are not pairwise isomorphic by 2. By 3.8(2), the module N/NJ is cyclic. Therefore, N = X + NJ, where X is a cyclic module. By the Nakayama lemma, N = X is a cyclic module and M is a Bezout module. 4. By 3, M is a finite direct sum of quotient finite-dimensional right A-modules. In [8, Corollary 5.7], it is proved that any finite direct sum of quotient finite-dimensional modules is a quotient finite-dimensional module; also see [7, Corollary 5.24]. ▷

Lemma 3.10. Let B be a ring containing some countable set {e(i)}∞ i=1 of non-zero central pairwise orthogonal idempotents e(i) such that all rings A(i) = e(i)A are domains and there exists an element b ∈ B such that e(i)b is a non-invertible non-zero element of the domain A(i) for every i. If φ is an automorphism of the ring B such that φ(e(i)) = e(i) for all i, then the ring B((x, φ)) is not a right Bezout ring and is not a right distributive ring. ◁ Let R = B((x, φ)) and C = ⨁∞ i=1 e(i)B. Then C is an ideal of the ring B. We denote by D the ideal of the ring B consisting of all elements d such that only finite number of projections de(i) are non-zero. It is clear that C ⊆ D and the element b is not contained in the ideal D, since be(i) ≠ 0 for all i. Since φ(e(i)) = e, the restriction of the automorphism φ to the ring A(i) is an automorphism of the ring A(i). We cannot distinguish the rings R(i) = e(i)R with skew Laurent series rings A(i)((x, φi )) over A(i). For every element a of the ring A, we denote by a(i) its projection aei onto the ring A(i). We consider elements f = e(1)x + e(2)x2 + ⋅ ⋅ ⋅ and b = bx0 of the series ring R. We assume that R is a right Bezout ring. Then there exists a series g with fR + bR = gR. It follows from this relation that all coefficients of the series g are contained in the right ideal C + bB of the ring B generated by coefficients series f and b. On the other hand, the element b also is contained in the right ideal of the ring B generated by coefficients of the series g. Therefore, all coefficients of the series g cannot be contained in the ideal D. We choose the lowest coefficient among of coefficients of the series g not contained in D. Let gi be this coefficient. All coefficients of the series g, which are lower gi , are contained in the ideal D. Since the number of these coefficients is finite, there exists a positive integer k such that projections of these coefficients onto A(n) are equal to zero for all n exceeding k. Since the coefficient gi is contained in the right ideal C+bA and is not contained in the ideal D, there exists an integer n > k such that the element

3.11 Bezout rings and principal right ideal rings | 25

gi (n) is contained in the right ideal bA(n) and it is not equal to zero. Then we consider the projection of the relation gR = fR + bR onto the ring R(n) = e(n)R. We obtain the relation e(n)gR = g(n)R(n) = f (n)R(n) + b(n)R(n)

= e(n)xn R(n) + b(n)R(n) = R(n) + b(n)R(n) = R(n).

Therefore, the series g(n) = e(n)g is invertible in the ring R(n). The element gi (n) ∈ A(n) is the lowest coefficient of the series g(n), since all coefficients of lower degrees are equal to zero (by the property that n > k). In addition, the element gi (n) is contained in the proper right ideal bA(n) = e(n)bA(n) of the ring A(n); consequently, it is not invertible in the ring A(n). By the property that the lowest term of the product is equal to the product of lowest terms (since A(n) is a domain), the series g(n) is not invertible in the ring R(n). This is a contradiction; thus, B((x, φ)) is not a right Bezout ring. Now we assume that the ring R is right distributive. By 3.9(1), for elements b and f of the right distributive ring R, there exists an element g of the ring R such that bg ∈ fR and f (1 − g) ∈ bR. It is directly verified that all coefficients of all series from fR are contained in the right ideal C which is generated by coefficients of the series f . Then all coefficients of the series bg are contained in C. In particular, for every i, only finite number of projections e(n)bgi = b(n)gi (n) is non-zero. Since each of the rings A(n) are domains and projections b(n) are non-zero for all n, we find that, for every i, only a finite number of projections gi (n) are non-zero. Therefore, all coefficients gi are contained in D. Let the lowest term of the series g be equal to g−t x −t . Since the subscript set {i} with −t ≤ i ≤ 0 is finite and all coefficients gi are contained in D, there exists a positive integer n such that all projections gi (n) are equal to zero for −t ≤ i ≤ 0. We multiply the inclusion f (1 − g) ∈ bR by the central idempotent e(n) and obtain the inclusion e(n)f (1 − g) ∈ b(n)R. Therefore, e(n)xn (1 − g) ∈ b(n)R(n). Since A(n) is a domain, an arbitrary series from the ring R(n) is invertible if and only if its lowest coefficient is invertible. By the choice of n, the lowest term of the series e(n)(1 − g) is equal to e(n). Consequently, the series e(n)xn (1 − g) is invertible in the ring R(n). Then the element b(n) is invertible in the ring R(n). Then the element b(n) is invertible in the ring A(n). This is a contradiction. ▷

3.11 Bezout rings and principal right ideal rings 1.

If A is a ring with automorphism φ and A is a principal right ideal ring, then A((x, φ)) is a principal right ideal ring, the Jacobson radical of the ring A((x, φ)) is nilpotent and coincides with N((x, φ)), where N is the prime radical of the ring A. 2. A ring A with automorphism φ is a principal right ideal domain if and only if A((x, φ)) is a principal right ideal domain. Under these conditions, A((x, φ)) is a semiprimitive domain.

26 | 3 Serial and Bezout rings A((x, φ)) 3.

If A is a ring with automorphism φ and A((x, φ)) is a semilocal right Bezout ring, then A((x, φ)) is a principal right ideal ring. 4. If A is a right distributive Bezout domain which is not a division ring (for example, the ring of integers) and B = A(1) × A(2) × A(3) × ⋅ ⋅ ⋅ is the direct product of a countable number of copies A(i) of the domain A, then B and B((x)) are reduced rings, in every of which the right annihilator of every element is generated by a central idempotent and coincides with the left annihilator, B is a right distributive right Bezout ring and the ring B((x)) is neither a right Bezout ring nor a right distributive ring. ◁ We denote R = A((x, φ)). 1. The assertion follows from Lemma 2.6(8) and Theorem 3.7. 2. If A is a principal right ideal domain, then R is a principal right ideal domain by 1 and 1.2(6). Now let R be a principal right ideal domain. Then its subring A is a domain as well. It follows from Lemma 1.4(1) that the domain R is semiprimitive. Now let B be an arbitrary non-zero right ideal of the domain A. The right ideal B((x, φ)) of the domain R is principal; it is generated by some non-zero series f . The series f can be represented as uxk , where k ∈ ℤ and the series u is contained in A[[x, φ]] and has the non-zero constant term b. Since x k is an invertible element of the domain R, we have gR = fR = B((x, φ)) and b ∈ B. Let b󸀠 be an arbitrary element of B. Then b󸀠 =∈ B((x, φ)) = gR. Therefore, b󸀠 = gh for some series h from the domain R. Since b = g0 and R is a domain, we have b󸀠 = bh0 , where h0 is the constant term of the series h ∈ R and h0 ∈ A. Therefore, B = bA is a principal right ideal of the domain A. 3. By 3.8(6), R is a right Bezout ring and all cyclic right R-modules are finitedimensional. By Theorem 3.7, the right Bezout ring R is a principal right ideal ring and its the Jacobson radical is nilpotent and coincides with N((x, φ)), where N is the prime radical of the ring A. 4. Since any direct product of reduced rings is a reduced ring, B is a reduced ring. Since B is a reduced ring, B((x)) is a reduced ring. For every element b of the ring B, its right annihilator coincides with a left annihilator and it is generated by a central idempotent e, where the components of the idempotent e are defined as follows: e(i) is equal to the identity element of the domain A(i) for i such that b(i) = 0 and e(i) is equal to zero for all the remaining i. For every positive integer i, we denote by e(i) the identity element of the ring A(i). In the ring B((x)), the right annihilator of any its element f coincides with its left annihilator and it is generated by a central idempotent t of the ring B, where components of the idempotent t are defined as follows: t(i) is equal to the identity element of the domain A(i) for i such that fe(i) = 0 and t(i) is equal to zero for all the remaining i.

3.13 See [71]

| 27

To prove that B is a right Bezout ring, it is sufficient show that any 2-generated right ideal is a principal right ideal. Let u, v be two arbitrary elements of the ring B and let u(i), v(i) be the projections of elements u and v onto the component A(i). For every i, there exists an element w(i) ∈ A such that w(i)A = u(i)A + v(i)A. For the element w of the ring B with components w(i), the relation wB = uB + vB holds, as required. To prove that B is a right distributive ring, it is sufficient to verify the distributivity criterion 3.9(1). Let u, v be arbitrary elements of the ring B and let u(i), v(i) be projections of the elements u, v onto the component A(i). Since A is distributive, for every i, there exists an element w(i) ∈ A such that u(i)w(i) ∈ v(i)A(i)

and v(i)(e(i) − w(i)) ∈ u(i)A(i).

For an element w ∈ B with components w(i), the relations uw ∈ v and v(1−w) ∈ uB hold, as required. The remaining part of the assertion 4 follows from Lemma 3.10 applied to the ring B, where we set φ = 1B . All components of the element b can be taken equal to a, where a is an arbitrary non-zero non-invertible element of the ring A. ▷ Theorem 3.12. Let A be a ring, φ its automorphism, and R the skew Laurent series ring A((x, φ)). The following conditions are equivalent. 1) R is a right serial ring. 2) R is a right serial right Artinian ring. 3) A is a right serial right Artinian ring. ◁ The equivalence of conditions 2) and 3) is proved in Proposition 3.2. The implication 2) ⇒ 1) is obvious. 1) ⇒ 2). Since every right series ring is semilocal, it follows from 3.9(4) that every cyclic right module over a right serial ring R is a quotient finite-dimensional module. By Theorem 3.7, R is a right Noetherian ring. By Proposition 3.2, R is a right Artinian right serial ring. ▷

3.13 See [71] Let A be a ring, a ∈ A. A right hollow chain of a is a strictly descending chain a0 A ⊋ a1 A ⊋ a2 A ⊋ a3 A ⊋ ⋅ ⋅ ⋅ , with a0 = a and for all n ≥ 0, an+1 = an − an bn an for some bn ∈ A. 1. See [71, Corollary 10] for the proof. The ring A is semilocal if and only if A does not have right hollow chains, a0 A ⊋ a1 A ⊋ a2 A ⊋ ⋅ ⋅ ⋅ ⊋ an A ⊋ ⋅ ⋅ ⋅

with ai+1 = ai − ai bi ai .

28 | 3 Serial and Bezout rings A((x, φ)) 2. If a, b are two elements of A such that a ∈ (1 − ab)A, then a is a right invertible element of A. Proof. Indeed, since abA + (1 − ab)A = A

and ab ∈ (1 − ab)A,

we have abA = A; cf. [71, Lemma 3]. Consequently, if the ring A is semilocal, then a is an invertible element. Theorem 3.14 ([72]). If R is a ring such that the Laurent series ring R((x)) is semilocal, then R is a semiperfect ring and its Jacobson radical is a nil-ideal of the ring R. Proof. We set A = R((x)). Assume that the ring A is semilocal. By Lemma 2.6(4), the ring R is semilocal. It is well known that, if we show that the Jacobson radical J(R) of the ring R is a nil-ideal, then idempotents of R/J(R) can be lifted modulo J(R) what together with the fact that R is semilocal will finish our proof. To show that J(R) is nil, consider an element r ∈ J(R), and assume for a contradiction that r ℓ ≠ 0 for every positive integer ℓ. Consider the following sequence of elements of A: a0 = rx−1 ,

an+1 = an − an rx−1 an

for all n ≥ 0.

Since A is semilocal, it follows from 3.13(1) that there exists n0 ∈ ℕ with an0 A = an0 +1 A. Hence an0 ∈ (1 − an0 rx−1 )A, and thus, by 3.13(2), 1 − an0 rx−1 is an invertible element of A. We claim that, for any n ≥ 1, the element 1 − an rx−1 is of the form 2n+1 −1

n+1

n+1

1 − an rx −1 = 1 + ∑ ki r i x i−1 + r 2 x −2 ,

(∗)

i=1

where ki ∈ ℤ for i ∈ {1, 2, . . . , 2n−1 }. Indeed, 1 − a1 rx−1 = 1 − r 2 x −2 + r 4 x −4 , and assuming (∗), we obtain 2

1 − an+1 rx−1 = 1 − (an − an rx−1 an )rx−1 = 1 − an rx−1 + (an rx−1 ) 2n+1 −1

n+1 −1 2

−1 i

= 1 + ∑ ki (rx ) + (rx ) i=1

2n+1 −1

−1 i

n+1 −1 2

+ ( ∑ ki (rx ) + (rx ) i=1

2

).

Thus our claim follows by induction. Hence f = 1 − an0 rx−1 is an invertible element of the form 2m −1

m

m

f = k0 ⋅ 1 + ∑ ki r i x −i + k2m r 2 x −2 , i=1

where m = n0 + 1 and k0 = k2m = 1.

3.13 See [71]

| 29

Since f ∈ U(A), there exists g ∈ A such that fg = 1. Let g = ∑j=q bj x j for some q ∈ ℤ. Notice that for every i ≠ 0 the coefficient of xi in f belongs to the right ideal rR ⊆ J(R). Thus looking at the coefficient of the x0 in fg, we have 2m

1 = 1 ⋅ b0 + ∑ ki r i bi = b0 + ra i=1

m

for some a ∈ R. So b0 = 1 − ra ∈ U(R), since r ∈ J(R). Now for the coefficient of x −2 in fg, we get 2m

2m −1

0 = r b0 + ∑ ki r i b−(2m −1) + k0 r 0 b−2m . i=1

m

Since b0 ∈ U(R), by our assumption about r, we have r 2 ⋅ b0 ≠ 0. Moreover, obvim ously rh ⋅ r 2 ⋅ b0 ≠ 0 for every positive integer h. Hence there exists d < 2m such that kd r d b−(2m −d) ≠ 0 and r h ⋅ kd r d b−(2m −d) ≠ 0 for every h, which together with the fact that kd ∈ ℤ obviously gives r h r d b−(2m −d) ≠ 0

for every h.

Since 2m − d > 0, we deduce that we can find the largest positive integer c such that r e ⋅ b−c ≠ 0 m

for every e. Now looking at the coefficient of the x−(2

+c)

, we have

2m −1

m

0 = r 2 b−c + ∑ ki r i b−(2m +c−i) + k0 r 0 b−(2m +c) . i=1

It is not hard to see that, by the assumption on c, we can find a positive integer v such that v

2m −1

r ⋅ ( ∑ ki r i b−(2m +c−i) + k0 r 0 b−(2m +c) ) = 0. i=1

But then v

2m

2m −1

m

0 = r ⋅ (r b−c + ∑ ki r i b−(2m +c−i) + k0 r 0 b(−2m +c) ) = r v r 2 b−c ≠ 0, i=1

a contradiction. Thus r has to be a nilpotent element of R.

30 | 3 Serial and Bezout rings A((x, φ))

3.15 Open questions Let R be a ring with automorphism φ. 1. If the skew Laurent series ring R((x, φ)) is semilocal, then is it true that the Jacobson radical J(R) is a nil-ideal of the ring R? 2. In terms of the ring R, can one find a criterion of the property that the ring R((x)) is semilocal? 3. In terms of the ring R and the automorphism φ, can one find a criterion of the property that the ring R((x, φ)) is semilocal? 4. Is it true that R((x, φ)) is a right semidistributive semilocal ring if and only if R is a right Artinian right semidistributive ring? 5. Find conditions on the ring R and the automorphism φ that are equivalent to the property that R((x, φ)) is a local ring. 6. Find conditions on the ring R and the automorphism φ that are equivalent to the property that R((x, φ)) is a semiperfect ring.

4 Prime and semiprime skew Laurent series rings In 4.1, we will recall some well known definitions and assertions which are used without special references and are directly verified; for example, see [57].

4.1 Some classes of modules and rings 1.

A submodule N of the module M is said to be fully invariant if f (N) ⊆ N for any endomorphism f of the module M. A module is said to be invariant (resp., quasiinvariant), if all its submodules (resp., maximal submodules) are fully invariant. Therefore, right invariant (resp., right quasi-invariant) rings coincide with the rings in which all right ideals (resp., maximal right ideals) are ideals. 2. A ring is said to be normal or Abelian if all its idempotents are central. Every right or left invariant ring is normal. The endomorphism ring of an invariant module is a normal ring. 3. A ring is called a prime ring (resp., a domain) if the product of any two its non-zero ideals (resp., elements) is not equal to zero. A ring is said to be semiprime (resp., reduced) if the square of every its non-zero ideal (resp., element) is not equal to zero. All prime rings (resp., domains) and subdirect products of semiprime (resp., reduced) rings are semiprime (resp., reduced) rings. 4. A ring A is said to be simple if any its non-zero ideal coincides with A. All division rings are simple rings and all matrix rings over a simple ring are simple rings.

4.2 Let A be a ring with automorphism φ and R = A((x, φ)). The definition of the λ(F) designation is given at the beginning of Section 1.2. 1. R = ((φ−1 , x))A and A[x, φ] = [φ−1 , x]A. If B and C are any two right ideals in A, then (B + C)((x, φ)) = B((x, φ)) + C((x, φ)) and B((x, φ)) ∩ C((x, φ)) = (B ∩ C)((x, φ)). Therefore, the right ideal lattice of the ring A is isomorphic to a sublattice of the right ideal lattice in R. 2. If B is a right ideal in A, then B((x, φ)) is a right ideal in R. In addition, if Bn = 0 and φ(B) ⊆ B, then (B((x, φ)))n = 0; in particular, B((x, φ)) ⊆ J(R). 3. Let B be an ideal of A. Then B((x, φ)) is an ideal in R if and only if φ(B) ⊆ B. In addition, if φ(B) = B, then B((x, φ)) is an ideal in R and the factor ring R/B((x, φ)) is isomorphic to the skew Laurent series ring (A/B)((x, φ)), where φ is the automorphism of the ring A/B induced by the automorphism φ of the ring A. https://doi.org/10.1515/9783110702248-004

32 | 4 Prime and semiprime skew Laurent series rings 4. If B and C are two ideals in A such that φ(B) ⊆ B,

φ(C) ⊆ C,

BC = 0,

then B((x, φ)) and C((x, φ)) are ideals in R and B((x, φ))C((x, φ)) = 0. 5. If F is a right ideal in R, then λ(F) is a right ideal in A and λ(F) coincides with set of constant terms of all series from F ∩ A[[x, φ]]. 6. If A is the finite direct product of rings A1 , . . . , An with identity elements e1 , . . . , en and φ(ei ) = ei for all i, then, for every i, the automorphism φ induces the automorphism φi of the ring Ai and R ≅ A1 ((x, φ1 )) × ⋅ ⋅ ⋅ × An ((x, φn )). 7.

If F is a non-zero ideal in R, then λ(F) is a non-zero ideal in A, λ(F) = φ(λ(F)),

λ(F)λ(G) ⊆ λ(FG)

for every ideal G in R. 8. If an element a ∈ A is right (resp., left) invertible in R, then a is right (resp., left) invertible in A. i If f = ∑∞ i=m ai x is a series in R such that ai ∈ A and the lowest coefficient am is right invertible (resp., left) in A, then the series f is right (resp., left) invertible in R, and if f is of degree 0 (i. e., m = 0), then the right (resp., left) inverse series for f is of degree 0 as well. 9. If F is a right ideal in R and λ(F) = A, then F = R. 10. If M is a maximal right ideal in A, then M((x, φ)) is a maximal right ideal in R. ◁ The assertions 1–6 are directly verified. 7. Without loss of generality, we can assume that F ≠ 0 and λ(F) is a non-zero ideal in A. Let 0 ≠ a ∈ λ(F). There exists a series f ∈ F ∩ A[[x, φ]] such that a is the constant term of the series f . Since F is an ideal in R, we have xf ∈ F and x −1 f ∈ F. In addition, φ(a) = λ(xf ) and φ−1 (a) = λ(x −1 f ). Therefore, φ(λ(F)) ⊆ λ(F),

φ−1 (λ(F)) ⊆ λ(F).

8. The first assertion is directly verified. We prove the second assertion. We assume that am a = 1 for some a ∈ A. For every integer j ≥ 0, we set gj = am+j φj (a) ∈ A. In addition, we set g = fx −m a = i−m j a = ∑∞ ∑∞ i=m ai x j=0 gj x ∈ A[[x, φ]] ⊆ R. Then g0 = 1. By 1.1(3), the series g is invertible in A[[x, φ]] ⊆ R. Then fx−m ag −1 = gg −1 = 1. Therefore, the series f is right invertible in R and if f is of degree 0, then the right inverse series for f also is of degree 0. The case of left invertible series can be considered similarly.

4.3

| 33

9.

By 5, λ(F) is a right ideal in A and there exists a series f ∈ F with lowest term 1. By 8, f ∈ U(R). Therefore, F = A. 10. It is sufficient to prove that, for every series t ∈ R \ M((x, φ)), there exist h ∈ M((x, φ)) and g ∈ R with h + tg = 1. Without loss of generality, we can assume i that t = ∑∞ i=0 ti x , where 0 ≠ t0 ∈ A \ M and ti ∈ A for all i. Since M is a maximal right ideal of A, we have m0 + t0 a0 = 1 for some m0 ∈ M and a0 ∈ A. Then ∞

m0 + ta0 = 1 + ∑ ti φi (a0 )x i . i=1

Therefore, (m0 + ta0 )f = 1 for some f ∈ R. We denote h = m0 f ∈ M and g = a0 f ∈ R. Then h + tg = 1. ▷

4.3 Let A be a ring with automorphism φ and R = A((x, φ)). i 1. J(R) ⊆ (J(A))((x, φ)) (i. e., for any series f = ∑∞ i=m fi x ∈ J(R), fi ∈ A, all coefficients fi are contained in J(A)), φ(J(A)) = J(A) and the ring R/(J(A))((x, φ)) is isomorphic to the skew Laurent series ring (A/J(A))((x, φ)), where φ is the automorphism of the ring A/J(A) induced by the automorphism φ of the ring A. Therefore, the rings R/(J(A))((x, φ)) and (A/J(A))((x, φ)) are isomorphic to the same factor ring of the ring R/J(R). In particular, if A is semiprimitive, then R is semiprimitive. 2. If R is semilocal, then A is semilocal. 3. If n ∈ ℕ and (J(A))n = 0, then J(R) = (J(A))((x, φ)),

n

(J(R)) = 0.

4. If R right finite-dimensional, then A is right finite-dimensional. 5. Let N be an ideal in A, φ(N) = N, φ be the automorphism A/N induced by the automorphism φ, and let the rings R, (A/N)((x, φ)) be right finite-dimensional. Then the two rings A and A/N are right finite-dimensional. 6. Let B be an ideal in A such that φ(B) = B and the skew Laurent series ring (A/B)((x, φ)) is semiprimitive, where φ is the automorphism of the ring A/B induced by the automorphism φ. Then B((x, φ)) ⊇ J(R). Therefore, if B((x, φ)) ⊆ J(R), then J(R) = B((x, φ)). 7. If N1 is the sum of all nilpotent ideals in A, then N1 ((x)) is an ideal in A((x)) contained in the sum of all nilpotent ideals of the ring A((x)); in particular, the ideal N1 ((x)) is contained in the Jacobson radical of the ring A((x)). 8. R is a division ring if and only if A is a division ring. 9. A((x, φ)) is a domain ⇔ A[x, x−1 , φ] is a domain ⇔ A is a domain.

34 | 4 Prime and semiprime skew Laurent series rings ◁ 1.

Let {Mi }i∈I be some set of maximal right ideals of A. By 4.2(10), Mi ((x, φ)) is a maximal right ideal of R for any i. Therefore, J(R) ⊆ ⋂i∈I (Mi ((x, φ))) = (J(A))((x, φ)). If M is a right ideal in A, then M ∈ max(AA )

⇔

φ(M) ∈ max AA

⇔

φ−1 (M) ∈ max AA .

Therefore, φ(J(A)) = J(A). By 4.2(3), R/(J(A))((x, φ)) ≅ (A/J(A))((x, φ)). 2. The assertion follows from 1 and the property that every factor ring of a semilocal ring is semilocal. 3. The assertion follows from the property that (J(A))((x, φ)) ⊆ J(R), by 4.2(2), and J(R) is contained in (J(A))((x, φ)) by 1. 4. The assertion follows from 4.2(1). 5. The assertion follows from 4 and 4.2(3). 6. Since the ring (A/B)((x, φ)) is semiprimitive and there exists a natural ring isomorphism R/(B((x, φ))) ≅ (A/B)((x, φ)), the ring R/(B((x, φ))) is semiprimitive. Therefore, B((x, φ)) ⊇ J(R). 7. The assertion follows from 4.2(2). 8. The assertion follows from 4.2(8). 9. If R is a domain, then its subring A[x, x−1 , φ] is a domain. If A[x, x −1 , φ] is a domain, then its subring A is a domain. If A is a domain, then the lowest coefficient of the product of any two non-zero series in R is not equal to zero. Therefore, R is a domain. ▷

4.4 Let A be a ring with automorphism φ, M a right A-module, and R ≡ A((x, φ)). 1. If M = ∑ni=1 mi A for some m1 , . . . , mn ∈ M, then M((x, φ)) = ∑ni=1 mi R. 2. If Y and Z are two distinct submodules in MA , then Y((x, φ)) ≠ Z((x, φ)),

(Y + Z)((x, φ)) = Y((x, φ)) + Z((x, φ)),

(Y ∩ Z)((x, φ)) = Y((x, φ)) ∩ Z((x, φ)).

3.

Therefore, the submodule lattice of MA is isomorphic to a sublattice of the submodule lattice of the right R-module M((x, φ)). N = λ(N((x, φ))) for every submodule N in MA .

4.4

| 35

4. The submodule lattice of MA is isomorphic to a sublattice of the submodule lattice of the right R-module M((x, φ)). In particular, if the R-module M((x, φ)) is Noetherian (resp., Artinian, distributive, uniserial), then the A-module M is Noetherian (resp., Artinian, distributive, uniserial). 5. For every series f ∈ M((x, φ)), there exists a series g ∈ M[[x, φ]] such that g has the lowest degree 0 and f A = gA. 6. Let F, G be two submodules of the right R-module M((x, φ)), F ⊆ G, λ(F) = λ(G), and let the right A-module λ(G) be finitely generated. Then F = G. 7. If MA is a Noetherian module, then there exists a mapping λ from the set of all submodules of the right R-module M((x, φ)) into the set of all submodules of the A-module M such that λ preserves proper inclusions; in particular, M((x, φ)) is a Noetherian R-module. 8. M is an Artinian Noetherian A-module if and only if M((x, φ)) is an Artinian Noetherian R-module. In this case, the composition length dA of the module MA is equal to the length dR of the module M((x, φ))R . 9. M is a simple A-module if and only if M((x, φ)) is a simple R-module. 10. If M is a finitely generated semisimple A-module, then M((x, φ)) is a finitely generated semisimple R-module. ◁ 1, 2, 3. The assertions are directly verified. 4. The assertion follows from 2. 5. Let t be the lowest degree of the series f and g ≡ fx −t . Since x −t ∈ U(R) and g has the lowest degree 0, we find that g is the required series. 6. We can assume that F ≠ 0. There exist non-zero elements f1 , . . . , fm ∈ F such that λ(F) = ∑m i=1 λ(fi )A. By 5, we can assume that all fi are of lowest degree 0. We have to prove that F contains all non-zero elements g ∈ G. By 5, we can assume that g is of lowest degree 0. We construct aij ∈ A (for i = 1, . . . , m and j = 0, 1, 2 . . .) such n j that, for all n = 0, 1, 2, . . . , g − ∑m i=1 ∑j=0 fi aij x has the lowest degree > n. Since m

λ(g) ∈ λ(G) = λ(F) = ∑ λ(fi )A, i=1

we have λ(g) = λ(f1 )a10 + ⋅ ⋅ ⋅ + λ(fm )am0 for some ai0 ∈ A. Since all f1 , . . . , fm , g have the lowest degree 0, we find that g − (f1 a10 + ⋅ ⋅ ⋅ + fm am0 ) has the lowest degree > 0. We assume that, for i = 1, . . . , m and j = 0, 1, . . . , n, we already have aij ∈ A such that n j h = g−∑m i=1 ∑j=0 fi aij x is of lowest degree > n. We note that h ∈ G. We denote by hn+1 the coefficient of xn+1 in h. We have either hn+1 = 0 or hn+1 = λ(h). Then hn+1 ∈ λ(G) in both cases. There exist elements ai,n+1 ∈ A such that hn+1 = λ(f1 )a1,n+1 + ⋅ ⋅ ⋅ + λ(fm )am,n+1 and hn+1 − (f1 a1,n+1 xn+1 + ⋅ ⋅ ⋅ + fm am,n+1 x n+1 ) has the lowest degree > n + 1.

36 | 4 Prime and semiprime skew Laurent series rings j This completes the induction step. Finally, we set di = ∑∞ j=0 aij x for all i = 1, . . . , m and obtain g = f1 d1 + ⋅ ⋅ ⋅ + fm dm . Therefore, g ∈ F. ▷ 7. The assertion follows from 6. 8. By 7 and 4, M is an Artinian Noetherian A-module if and only if M((x, φ)) is an Artinian Noetherian R-module. By 4, dA ≤ dR . By 7, dR ≤ dA . 9. The assertion follows from 7 and 4. 10. Let M = ∑ni=1 mi A, where all A-modules mi A are simple. By 4.2, M((x, φ)) = ∑ni=1 mi R. By 9, all R-modules mi R are simple. Therefore, M((x, φ)) is a finitely generated semisimple R-module. ▷

Proposition 4.7. If A is a ring with automorphism φ, then the following conditions are equivalent. 1) A((x, φ)) is a simple ring. 2) φ(B) ⊈ B for any non-zero proper ideal B in A. 3) φ(B) ≠ B for any non-zero proper ideal B in A. ◁ We set R = A((x, φ)). The implication 2) ⇒ 3) is obvious. 3) ⇒ 1). Let F be a non-zero ideal in R. Then λ(F) is a non-zero ideal in A and λ(F) = φ(λ(F)). By 3), λ(F) = A. By 4.2(9), F = R. 1) ⇒ 2). We assume the contrary. By 4.2(3), B((x, φ)) is a non-zero ideal in R. Since R is a simple ring, B((x, φ)) = R. Therefore, B = A. This is a contradiction. ▷ Example 4.8. There exists a simple skew Laurent series ring such as with non-simple coefficient ring. Let A1 , A2 be two copies of the field F, A be the direct product of rings A1 and A2 , and let φ be an automorphism of the ring A such that φ(a1 , a2 ) = (a2 , a1 ) for all a1 ∈ A1 and a2 ∈ A2 . Then the ring A is not simple and the ring A((x, φ)) is simple by 4.2. ▷ Proposition 4.9. If A is a ring with automorphism φ of A, then the following conditions are equivalent. 1) A((x, φ)) is a prime ring. 2) A does not have non-zero ideals B and C such that BC = 0, φ(B) ⊆ B, and φ(C) ⊆ C. 3) A does not have non-zero ideals B and C such that BC = 0, φ(B) = B, and φ(C) = C. ◁ We set R = A((x, φ)). The implication 2) ⇒ 3) is obvious. 3) ⇒ 1). We assume that R has two non-zero ideals F and G with FG = 0. By 4.2(7), λ(F)λ(G) ⊆ λ(FG) = λ(0) = 0. By 4.2(7), φ(λ(F)) = λ(F) and φ(λ(G)) = λ(G). This is a contradiction. 1) ⇒ 2). We assume that A has two non-zero ideals B and C such that BC = 0,

φ(B) ⊆ B,

φ(C) ⊆ C.

4.11

| 37

By 4.2(4), B((x, φ)) and C((x, φ)) are ideals of the ring R and B((x, φ))C((x, φ)) = 0. This is a contradiction. ▷ Proposition 4.10. If A is a ring with automorphism φ, then the following conditions are equivalent. 1) A((x, φ)) is a semiprime ring. 2) φ(B) ⊈ B for any non-zero nilpotent ideal B in A. 3) A does not have a non-zero ideal B such that φ(B) = B and B2 = 0. ◁ We set R = A((x, φ)). The implication 2) ⇒ 3) is obvious. 3) ⇒ 1). We assume that R has a non-zero ideal F such that F n = 0 for some n ∈ ℕ. By 4.2(7), n

(λ(F)) ⊆ λ(F n ) = λ(0) = 0. In addition, φ(λ(F)) = λ(F) by 4.2(7). This is a contradiction. 1) ⇒ 2). We assume that A has a non-zero nilpotent ideal B with φ(B) ⊆ B. By 4.2(4), B((x, φ)) is an ideal in R and (B((x, φ)))n = 0. This is a contradiction. ▷

4.11 Let A be a ring with automorphism φ. 1. If A is a semiprime ring, then A((x, φ)) is a semiprime ring. 2. If A is a ring with maximum condition on nilpotent ideals, then A((x, φ)) is semiprime if and only if A is semiprime. ◁ 1. The assertion follows from 4.10. 2. The implication ⇐ follows from 1. ⇒. We assume the contrary. Then B2 = 0 for some non-zero ideal B in A. For every i ∈ ℤ, we have (φi (B))2 = 0 and (B + φ(B) + ⋅ ⋅ ⋅ + φi (B))i+2 = 0. Since A is a ring with maximum condition on nilpotent ideals, we have ∞

n

i=0

i=0

∑ φi (B) = ∑ φi (B) ≡ B

for some n ∈ ℕ. Then B is a non-zero nilpotent ideal in A and φ(B) ⊆ B. By 4.10, the ring A((x, φ)) is not semiprime; this is a contradiction. ▷ Example 4.12. There exists a ring A with automorphism α such that the ring A((x, φ)) is semiprime and A is not semiprime. Let F be a field, ℤ the set of all integers, {xi }i∈ℤ a countable set of variables, A ≡ F[{xi }]i∈ℤ the polynomial ring in all variables xi , A the factor ring of the ring A modulo

38 | 4 Prime and semiprime skew Laurent series rings the ideal generated formed elements xi2 , h: A → A the natural epimorphism, and L the set of all poly-linear polynomials over A. Then A = h(L) and A is not semiprime. Every element a ∈ A has the following form: a = h(∑ fi1 ,...,in xi1 . . . xin ) (fi1 ,...,in ∈ F and ij ≠ ik for j ≠ k). The relation α(h(∑ fi1 ,...,in xi1 . . . xin )) = h(∑ fi1 ,...,in xi1 +1 . . . xin +1 ) defines an automorphism α of the ring A. We assume that the ring A((x, α)) is not semiprime. By 4.10, A has a non-zero ideal B such that α(B) = B and B2 = 0. Let 0 ≠ b = h(∑ fi1 ,...,in xi1 . . . xin ) ∈ B. There exists a positive integer m such that αm (b) = h(∑ fi1 ,...,in xi1 +m . . . xin +m ) and ij ≠ ik + m for j, k = 1, . . . , n. Then bαm (b) ≠ 0. On the other hand, bαm (b) ∈ B2 = 0, since α(B) = B and B2 = 0. This is a contradiction. ▷ Proposition 4.13. Let A be a ring with automorphism φ. 1. A((x, φ)) is a right Noetherian semiprime ring if and only if A is a right Noetherian semiprime ring. 2. A((x, φ)) is a semisimple ring if and only if A is a semisimple ring. 3. A((x, φ)) is a prime Artinian ring if and only if A is a semisimple ring and A does not have non-zero ideals B and C such that BC = 0,

φ(B) = B,

φ(C) = C.

◁ We set R = A((x, φ)). 1. By 4.4(4) and 4.4(7), we can assume that R and A are right Noetherian rings. Therefore, 1 follows from 4.11(2). 2. The implication ⇐ follows from 4.4(10). ⇒. The semisimple ring R is a right Artinian semiprime ring. By 4.4(4), A is right Artinian. By 1, A is semiprime. The right Artinian semiprime ring A is semisimple. 3. By 2, we can assume that R and A are semisimple rings. Therefore, the assertion follows from 4.7. ▷ Example 4.14. There exists a prime Artinian skew Laurent series ring such that its coefficient ring is not a prime ring. Let B1 and B2 be two copies of the field F, B be the direct product of fields B1 and B2 , and let φ be an automorphism of the ring B such that φ(b1 , b2 ) = (b2 , b1 ) for all b1 ∈ B1 and b2 ∈ B2 . Then B is a semisimple non-prime ring and B((x, φ)) is a prime Artinian ring by 4.13(3). ▷

5 Regular and biregular Laurent series rings In 5.1, we gather some well-known definitions and assertions, which are used without special links and are directly verified; e. g., see [12, 58, 57].

5.1 Regular and strongly regular modules and rings 1.

A module M is said to be regular if it satisfies the following equivalent conditions. a) In M, every cyclic submodule is a direct summand. b) In M, every finitely generated submodule is a direct summand. c) For every finitely generated submodule X of the module M, there exists an idempotent e of the endomorphism ring End M with X = eM. Every semisimple module is regular. Direct products of an infinite number of division rings are regular non-semisimple modules over themselves. 2. A module M is said to be strongly regular if it satisfies the following equivalent conditions. a) In M, every cyclic submodule is a fully invariant direct summand. b) In M, every finitely generated submodule is a fully invariant direct summand. c) For every finitely generated submodule X of the module M, there exists a central idempotent e of the endomorphism ring End M with X = eM. 3. A ring A is said to be (von Neumann) regular if it satisfies the following equivalent conditions. a) The module AA (the module A A) is regular. b) In A, every principal right (left) ideal is generated by an idempotent. c) In A, every finitely generated right (left) ideal is generated by an idempotent. d) a ∈ aAa for any element a ∈ A. 4. A ring A is said to be strongly regular if it satisfies the following equivalent conditions. a) The module AA (module A A) strongly is regular. b) In A, every principal right (left) ideal is generated by a central idempotent. c) In A, every finitely generated right (left) ideal is generated by a central idempotent. d) a ∈ a2 A for any element a ∈ A. e) a ∈ Aa2 for any element a ∈ A. f) A is a regular normal ring. g) A is a regular distributive invariant ring. h) A is a right or left quasi-invariant regular ring. i) A is a regular right distributive or left ring. j) A is a regular reduced ring. k) every element of the ring A is a product of a central idempotent and an invertible element. https://doi.org/10.1515/9783110702248-005

40 | 5 Regular and biregular Laurent series rings

5.

For n ≥ 2, the ring of all n × n matrices over a division ring is regular but is not strongly regular. A ring is said to be biregular if every its two-sided ideal generated by one element, is generated by a central idempotent. Any finite direct product of simple of rings is a biregular ring. Every biregular ring is semiprime. All strongly regular rings are biregular. For n ≥ 2 the ring of all n × n matrices over division ring is biregular, but it is not strongly regular.

5.2 φ-reduced rings A ring A with automorphism φ is said to be φ-reduced if A is a reduced ring and AaA ∩ Aφ(a)A ≠ 0 for every non-zero a ∈ A. If φ is an automorphism of the ring A, then the following conditions are equivalent. 1) ((φ, x))A is a reduced ring; 2) A is a φ-reduced ring; 3) A is a reduced ring and X ∩ φ(X) ≠ 0, X ∩ φ−1 (X) ≠ 0 for every non-zero ideal X in A; 4) A is a reduced ring and B ∩ φn (C) ≠ 0 for every n ∈ ℤ and any two ideals B and C in A such that B ∩ C ≠ 0; 5) ((φ, x))A is a semiprimitive reduced ring and A is a φ-reduced ring. ◁ We denote R = ((φ, x))A. The implications 2) ⇒ 3) and 5) ⇒ 1) are directly verified. 1) ⇒ 2). Let a ∈ A and AaA ∩ Aφ(a)A = 0. Then φ(a)a ∈ AaA ∩ Aφ(a)A = 0

and (xa)2 = xaxa = x2 φ(a)a = 0.

Since the ring R is reduced, xa = 0 and a = 0. 3) ⇒ 4). It follows from 3) that 0 ≠ (B ∩ C) ∩ φ(B ∩ C) ⊆ B ∩ φ(C) and

0 ≠ (B ∩ C) ∩ φ−1 (B ∩ C) ⊆ B ∩ φ−1 (C).

We have proved 4) for the cases n = 1 and n = −1. We will use induction on |n|. Let k ∈ ℕ. We assume that condition 4) is true for all integers n with |n| ≤ k. Then B ∩ φk (C) ≠ 0 and B ∩ φ−k (C) ≠ 0. By condition 3), 0 ≠ (B ∩ φk (C)) ∩ φ(B ∩ φk (C)) ⊆ B ∩ φk+1 (C), 0 ≠ (B ∩ φ−k (C)) ∩ φ−1 (B ∩ φ−k (C)) ⊆ B ∩ φ−k−1 (C). i 4) ⇒ 1). We assume that f = ∑∞ i=n ai x ∈ R, ai ∈ A for all i and an ≠ 0. By condition 4), Aan A ∩ Aφn (an )A ≠ 0. Since A is a reduced ring, an φn (an ) ≠ 0. Therefore, 0 ≠ an φn (an ) is the canonical coefficient of the lowest term of the series f 2 , f 2 ≠ 0, and R is reduced.

5.3

|

41

1) + 2) + 3) + 4) ⇒ 5). By 1) and 2), the ring R is reduced and A is φ-reduced. We i assume that J(R) ≠ 0. There exists a series f = ∑∞ i=−1 fi x ∈ J(R) such that f−1 ≠ 0 and all fi are contained in A. Since f ∈ J(R), we have 1 − f ∈ U(R). We set g = (1 − f )−1 = j ∑∞ j=−m gj x ∈ R, where g−m ≠ 0 and all gj are contained in A. We consider the case m > 0. By comparing coefficients of x −m−1 , we obtain −1 f−1 φ (g−m ) = 0. Since the ring A is reduced, Af−1 A ∩ Aφ−1 (g−m )A = 0. By condition 3), g−m f−1 = 0. By comparing coefficients of x−m , we obtain f−1 φ−1 (g−m+1 ) + (1 − f0 )g−m = 0. By multiplying this relation by g−m and considering that g−m f−1 = 0, we obtain 0 = g−m f−1 φ−1 (g−m+1 ) + g−m (1 − f0 )g−m = g−m (1 − f0 )g−m . Then (g−m (1 − f0 ))2 = 0. Therefore, g−m (1 − f0 ) = 0. By 4.3(1) f0 ∈ J(A). Therefore, 1 − f0 ∈ U(A). Therefore, g−m = 0. This is a contradiction. We consider the case m = 0. Then f−1 φ−1 (g1 ) + (1 − f0 )g0 = 1. Therefore, g0 = (1 − f0 )−1 (1 − f−1 φ−1 (g1 )) ∈ U(A). Therefore, g −1 = 1 − f ∈ A[[x, φ]] and f−1 = 0. We consider the case m < 0. Then ∞

f = ∑ fi xi ∈ J(R), i=−1

f−1 ≠ 0,

∞

g = (1 − f )−1 = ∑ gj x j . j=−m

Therefore, m = −1 and f−1 φ−1 (g1 ) = 1. By 4.3(1), f−1 ∈ J(A). This is a contradiction. ▷

5.3 Let A be a ring with automorphism φ and R = A((x, φ)). 1. If e is a central idempotent in R and e ∈ A, then φ(e) = e. 2. The ring R does not have a non-zero idempotent f with the lowest degree m > 0. i 3. If e = ∑∞ i=m ei x is a non-zero central idempotent in R, where ei ∈ A and em ≠ 0, then either e ∈ A and φ(e) = e or A has non-zero nilpotent element a such that aA = Aa = AaA,

(AaA)2 = 0,

and φ(a) = a;

in particular, A is not semiprime. 4. If A is semiprime, then every central idempotent e of the ring R is contained in A and φ(e) = e. 5. If A contains an infinite set {ei | 0 ≤ i < ∞} of non-zero orthogonal idempotents ∞ k i i such that φi (ei ) ∉ ∑i−1 k=0 φ (ek )A for all i, then, for the series z ≡ ∑i=0 φ (ei )x ∈ A((x, φ)), there does not exist an idempotent f ∈ R with zR = fR.

42 | 5 Regular and biregular Laurent series rings 6. If A contains an infinite set {ei | 0 ≤ i < ∞} of non-zero central orthogonal idempotents such that φ(ei ) = ei for all i, then the two-sided ideal in R, which is geni erated by the series z ≡ ∑∞ i=0 ei x , is not generated by a central idempotent of the ring R. ◁ 1. The assertion follows from the property that ex = xe = φ(e)x. i 2. We assume that there exists an idempotent f = ∑∞ i=m fi x ∈ R such that m > 0, fi ∈ A and fm ≠ 0. Then f = (fm + gx)xm , where g ∈ A[[x, φ]]. Therefore, (fm + gx)xm = f = f 2 = (fm + gx)x m (fm + gx)x m . 3.

Therefore, fm + gx = (fm + gx)xm (fm + gx). Then fm = 0; this is a contradiction. We consider the case m < 0. Since xe = ex, we have φ(em ) = em . In addition, 2 em φm (em ) = 0, since e2 = e. Therefore, em = 0. Since e is a central idempotent in m m R, we have bem x = em x b for all b ∈ A. Therefore, Aem = em A = Aem A. Now we can set a = em . The case m > 0 is impossible, by 2. We consider the case m = 0. Then e = e0 + ux, where e0 ∈ A. Since ex = xe, we find that e0 is a non-zero central idempotent in R, φ(e0 ) = e0 and u ∈ A[[x, φ]]. The element (1 − e0 )e = (1 − e0 )ux is a central idempotent in R. By 2, (1 − e0 )e = 0. Therefore, e = e0 e = e0 (e0 + ux) = e0 (1 + ux).

By 4.2(8), 1 + ux ∈ U(R). Therefore, e0 (1 + ux) = e = e2 = e0 (1 + ux)2 . Therefore, e = e0 (1 + ux) = e0 ∈ A. 4. The assertion follows from 3. j 5. We assume that there exists an idempotent f = ∑∞ j=t fj x (t ≤ 0) with fR = zR. Then fz = z. By comparing coefficients of xi , we obtain

ft φi (ei−t ) + ⋅ ⋅ ⋅ + f0 φi (ei ) + ⋅ ⋅ ⋅ + fi φi (e0 ) = φi (ei )

(0 ≤ i < ∞).

Since ei are orthogonal idempotents, the idempotents φi (e0 ), φi (e1 ), . . . are orthogonal for every fixed subscript i. Therefore, f0 φi (ei ) = φi (ei ). In addition, f = zh for some h ∈ A. Therefore, f0 = ∑rk=0 φk (ek )hk for some h0 , . . . , hr ∈ A. Then r

r

k=0

k=0

φr+1 (er+1 ) = f0 φr+1 (er+1 ) = ∑ φr (ek )hk φr+1 (er+1 ) ∈ ∑ φk (ek )A; this contradicts the assumption. 6. Since φ(ei ) = ei , the series z is contained in the center of the ring R. Therefore, the ideal in R, generated by z, coincides with zR. If zR = fR for some central idempotent f , we have the contradiction to 5. ▷

5.5

| 43

Theorem 5.4 ([54]). If A is a ring with automorphism φ, then the following conditions are equivalent. 1) A((x, φ)) is a biregular ring. 2) A((x, φ)) is the finite direct product of simple rings R1 , . . . , Rn . 3) A is the finite direct product of rings A1 , . . . , An with identity elements e1 , . . . , en such that φ(ei ) = ei for all i and every ring Ai coincides with any its non-zero ideal B such that φ(B) = B. ◁ Let R = A((x, φ)). The implication 2) ⇒ 1) is directly verified. 1) ⇒ 3). We prove that the ring A is biregular. Let 0 ≠ b ∈ A. There exists a non-zero central idempotent e of the biregular ring R with RbR = eR. If e ∈ A, then AbA = eA and A is biregular. We assume that e ∉ A. By 5.3(3), there exists a non-zero nilpotent element a ∈ A such that aA = Aa = AaA,

(AaA)2 = 0,

φ(a) = a.

Since φ(a) = a and (AaA)2 = 0, we find that (AaA)((x, φ)) is a non-zero nilpotent ideal of the biregular ring R; this is impossible. Therefore, A is biregular. In particular, A is semiprime. By 5.3(4) and 5.3(6), the biregular ring R does not contain an infinite set of central orthogonal idempotents. Then R is the finite direct product of simple rings R1 , . . . , Rn . Let ei be the identity element of the simple ring Ri and Ai ≡ ei A. By 5.3(4), all idempotents ei are contained in A and φ(ei ) = ei for all i. Therefore, A is the direct product of rings Ai , φ induces automorphisms φi of rings Ai , Ri ≅ Ai ((x, φi )) and A((x, φ)) ≅ A1 ((x, φ1 )) × ⋅ ⋅ ⋅ × An ((x, φn )). By 4.7, every ring Ai does not have a non-zero proper ideal B with φ(B) = B. 3) ⇒ 2). Since φ(ei ) = ei , the automorphism φ induces the automorphism φi of the ring Ai for every i and A((x, φ)) ≅ A1 ((x, φ1 )) × ⋅ ⋅ ⋅ × An ((x, φn )). By 4.7, each of the rings Ai ((x, φi )) are simple. ▷

5.5 If A is a ring with automorphism φ, then the following conditions are equivalent. 1) A((x, φ)) is a reduced ring and its factor ring modulo the Jacobson radical is regular. 2) A((x, φ)) is a regular normal ring. 3) A((x, φ)) is the finite direct product of division rings. 4) A is a finite the direct product of division rings and φ(e) = e for every central idempotent e ∈ A.

44 | 5 Regular and biregular Laurent series rings ◁ We set R = A((x, φ)). The implication 2) ⇒ 1) is directly verified. 1) ⇒ 2). Since R is reduced, J(R) = 0 by 5.2. Then R is regular, since R/J(R) is regular by assumption. The regular reduced ring R is strongly regular by 5.1(4). 2) ⇒ 3). The strongly regular ring R is biregular. By 5.4, R is the finite direct product of simple rings R1 , . . . , Rn . Every simple strongly regular ring Ri is a division ring. 3) ⇒ 4). Since R is a reduced ring, A is a reduced ring. By 5.3(4), φ(e) = e for every central idempotent e ∈ A. Since R is semisimple, A is semisimple by 4.13(2). It is clear that the reduced semisimple ring A is the finite direct product of division rings. 4) ⇒ 2). Since A is semisimple, R is semisimple by 4.13(2). With the use of 5.2 it is directly verified that R is reduced. Since the reduced semisimple ring is the finite direct product of division rings, R is strongly regular. ▷

5.6 Let A be a ring with automorphism φ. 1. If A is regular and has a proper essential right ideal B with φ(B) ⊆ B, then A contains an infinite set of non-zero orthogonal idempotents {ei | 0 ≤ i < ∞} such that k φi (ei ) ∉ ∑i−1 k=0 φ (ek )A for all i. 2. If A((x, φ)) is a regular non-semisimple ring, then A is a regular non-semisimple ring and φ(B) ⊈ B for any proper essential right ideal B in A. ◁ 1.

By the Zorn lemma, BA is an essential extension the direct sum of cyclic modules. Therefore, there exists a right ideal C = ⨁s∈S as A such that BA is an essential extension of CA . Then C is a proper essential right ideal in A and ∞

∑ φk (C) ⊆ B ≠ A.

k=0

We choose any b0 ∈ {as | s ∈ S}. We choose elements bi ∈ {as | s ∈ S} \ {b0 , . . . , bi−1 } such that bi ∉ Ci , where i

i−1

Ci ≡ ∑ ∑ φk (bj )A ≠ A. k=0 j=0

We will prove that it is possible to choose the elements bi . We assume that the choice of bm is impossible (i. e., φm (as )∈ Cm ) for every as ∈ {as | s ∈ S} \ {b0 , . . . , bm−1 }.

5.6

| 45

The essential right ideal C is contained in the proper finitely generated right ideal φ−m (Cm ); this contradicts the property that A is regular. By 4.7, there exist orthogonal idempotents ei (0 ≤ i < ∞) such that ⨁di=0 ei A = ⨁di=0 bi A for every d. We k assume that there exists a subscript i ≥ 1 such that φi (ei ) ∈ ∑i−1 k=0 φ (ek )A. Then i φ (ei ) ∈ Ci , since j

φj (ej ) ∈ φj ( ∑ bk A) ⊆ Ci , k=0

0 ≤ j ≤ i − 1.

Therefore, φi (bi ) ∈ Ci , since q

φi (eq ) ∈ φq ( ∑ bk A) ⊆ Ci k=0

(q ≤ i − 1);

this contradicts the choice of bi . 2. Since A((x, φ)) is not semisimple, A is not semisimple by 4.13(2). For every a ∈ A, i there exists a series f = ∑∞ i=t fi x with afa = a. Then af0 a = a and the ring A is regular. We assume that A has a proper essential right ideal B with φ(B) ⊆ B. By 5.6, there exists an infinite set {ei | 0 ≤ i < ∞} of orthogonal idempotents in A such that k φi (ei )∉∑i−1 k=0 φ (ek )A for all i. This contradicts 5.3(5). ▷ Proposition 5.7. If A is a ring with automorphism φ, then the following conditions are equivalent. 1) A((x, φ)) is a regular ring and Soc(AA ) is an essential right ideal in A. 2) A((x, φ)) is a semisimple ring. 3) A is a semisimple ring. ◁ The equivalence 2) ⇔ 3) is proved in 4.13(2). The implication 2)+3) ⇒ 1) follows from the property that any semisimple ring is regular. 1) ⇒ 3). We set R = A((x, φ)), B = Soc(AA ). We assume that A is not semisimple. Then B is a proper essential right ideal in A. By 4.13(2), R is not semisimple. Then A is a regular non-semisimple ring. By 5.6(2), φ(B) ⊈ B. This is a contradiction, since the socle of any ring is closed under automorphisms. ▷ Theorem 5.8 ([53]). If A is a ring with automorphism φ and φn is the identity automorphism of A for some n ∈ ℕ, then the following conditions are equivalent. 1) A((x, φ)) is a regular ring. 2) A is a semisimple ring. ◁ The implication 2) ⇒ 1) follows from 5.7. 1) ⇒ 2). We denote A((x, φ)) by R. We assume that A is not semisimple. By 4.13(2), R is not semisimple. By 5.6(2), A is a regular non-semisimple ring. Then A has a proper

46 | 5 Regular and biregular Laurent series rings essential right ideal C. We denote by B the proper essential right ideal ⋂nk=0 φk (C) of the ring A. Since φn ≡ 1A , we have φ(B) ⊆ B. This contradicts 5.6(2). ▷

5.9 Let A be a ring with automorphism φ and φ(e) = e for every central idempotent e ∈ A. 1. If A is a biregular reduced ring, then A((x, φ)) is a reduced ring and all its idempotents are contained in A. 2. If A is a strongly regular ring, then A((x, φ)) is a reduced ring such that all idempotents are contained in A and every non-zero right ideal contains a non-zero central idempotent. ◁ We set R = A((x, φ)). 1. Let 0 ≠ a ∈ A. Since A is biregular, AaA = eA for some central idempotent e ∈ A. By assumption, φ(e) = e. Therefore, Aφ(a)A = φ(eA) = eA = AaA. By 5.2, the ring R is reduced. Then the ring R is normal. By 5.3(4), all idempotents of the ring R are contained in A. 2. The strongly regular ring A is a biregular reduced ring. By 1, the ring R is reduced and all its idempotents are contained in A. We prove that every non-zero right i ideal F in R contains a non-zero central idempotent. Let f = ∑∞ i=m fi x be a nonzero series in F, where 0 ≠ fm ∈ A and fi ∈ A for all i. Since A is strongly regular, fm = ev for some non-zero central idempotent e ∈ A and v ∈ U(A). Since φ(e) = e, i the idempotent e is central in R. Therefore, fe = e ∑∞ i=m efi x , where efm = ev is an invertible element of the ring eA. Therefore, fe is an invertible element of the ring eR with the inverse element eg. Then e = feg is a non-zero central idempotent and e ∈ F. ▷

6 Equivalent definitions of Laurent rings The construction the ordinary Laurent series ring allows for various generalizations. In this section, we will give two equivalent definitions of Laurent rings. Skew Laurent series rings and pseudo-differential operator rings are partial cases of Laurent rings.

6.1 Pseudo-differential operator rings A((t −1 , δ)) Let A be a ring with derivation δ; i. e., δ is an endomorphism of the additive Abelian group A+ which satisfies the relation δ(ab) = δ(a)b+aδ(b). We denote by A((t −1 , δ)) the pseudo-differential operator ring over the coefficient ring A consisting of the formal series m

f = ∑ fi t i , i=−∞

where t is a variable, m is an integer (maybe, negative) and the coefficients fi of the series f are elements of the ring A. In the ring A((t −1 , δ)), addition is defined as usual and multiplication is defined by considering the rules ta = at + δ(a),

+∞

t −1 a = ∑ (−1)i δi (a)t −i−1 . i=0

The check of the property that the set A((t −1 , δ)) satisfies all ring axioms will be given below in Proposition 9.7. In the case where δ is the zero derivation, there exists an isomorphism from the ring A((t −1 , δ)) onto the ordinary Laurent series ring A((x)) (this isomorphism maps t −1 to x). It follows from the definitions of the skew Laurent series ring A((x, φ)) and the pseudo-differential operator ring A((t −1 , δ)) over the same coefficient ring A that there exists a natural bijection between these two rings; it maps x to t −1 and a formal sum of degrees x to the corresponding formal sum of degrees t −1 . This bijection is an isomorphism of left modules A A((x, φ)) and A A((t −1 , δ)) over the ring A. This similarity of these two structures gives them close ring properties and makes it possible to prove, in some cases, very similar theorems for rings of skew Laurent series and rings of pseudodifferential operators, and the proofs are also very similar. In connection to this, it is convenient to define Laurent rings, since the skew Laurent series rings and pseudo-differential operator rings are particular cases of Laurent rings; we will prove many theorems in such generality. In 6.2 and 6.3, two equivalent definitions of a Laurent ring are given. The equivalence of these definitions is proved in 6.5. https://doi.org/10.1515/9783110702248-006

48 | 6 Equivalent definitions of Laurent rings

6.2 The first definition of a Laurent ring A ring R is called a Laurent ring with coefficient ring A if there exists an isomorphism π from the Abelian additive group of the Laurent series ring A+ ((x)) onto the additive Abelian group R+ and π satisfied the properties 1)–4). 1) The restriction mapping π to the ring A is a unitary ring monomorphism from the ring A into the ring R. 2) π carries out isomorphism from the left module A A((x)) onto the left module A R, where the module A R is defined in correspondence to the relation ar = π(a)r. 3) The lowest degree of the product of two series is not less than the sum of their lowest degrees. 4) The restriction of the mapping π to the multiplicative group, generated by the element x, is a group monomorphism. Properties a, b of the mapping π Let R be a Laurent ring with mapping π from A((x)) into R. Then the above mapping π has the following properties a and b. a. For any series f in A((x)) and any element a of the ring A, the relation π(a)π(f ) = π(af ) holds. b. For any series f in A((x)) and any integer n, the relation π(f )π(x n ) = π(fxn ) holds. ◁ It follows from the condition 2) that, for any two elements a and b from the ring A and each integer n, the relation π(a)π(bxm ) = π(abxm ) holds; in addition, it follows from the condition 4) that, for any element a of the ring A and for any integers n and m, the relation π(bxm )π(xn ) = π(bxn+m ) holds. Therefore, the relations π(a)π(f ) = π(af ) and π(f )π(x n ) = π(fxn ) have been proved in the cases, where f has the form bxm . Since multiplication is distributive, these relations are extended to the Laurent polynomial ring A[x, x −1 ]. For any series f in A((x)) and any integer m, we can find a Laurent polynomial f 󸀠 such that f − f 󸀠 ∈ Vm ; therefore π(a)π(f ) − π(af ) = π(a)π(f − f 󸀠 ) − π(a(f − f 󸀠 )) ∈ π(Vm ). Since m is arbitrary, we obtain π(a)π(f ) − π(af ) = 0; this means that the relation π(a)π(f ) = π(af ) is extended to all series f in A((x)). Similarly, we find that the relation π(f )π(xn ) = π(fx n ) is extended to all series f from A((x)) ▷ Reformulation of the first definition of a Laurent ring If we identify every series f from the Laurent series ring A((x)) with the corresponding element π(f ) of the Laurent ring, then it follows from the proved properties a and b that we can give the first definition of a Laurent ring which can be formulated as follows:

6.3 The second definition of a Laurent ring

| 49

the additive Abelian group A+ ((x)) with multiplication ∘ is called a Laurent ring if it is a ring satisfying the relation (af ) ∘ (gxn ) = a(f ∘ g)x n , where a ∈ A; in addition, the lowest degree of the product of two series is not less than sum of their lowest degrees. It is clear that the ring A((x)) is a Laurent ring with π = 1A((x)) . Now we give the second definition of a Laurent ring. In 6.5 below, the equivalence of two definitions of a Laurent ring will be proved.

6.3 The second definition of a Laurent ring A ring R is called a Laurent ring if R contains a family of additive subgroups {Ui | − ∞ < i < +∞} which satisfies the following properties. i) For any two integers n and k, there are inclusions Un+1 ⊆ Un and Un Uk ⊆ Un+k ; in particular, this implies that U0 is a subring in R and U1 is a two-sided ideal in U0 . In addition, the union Un by all integers n coincides with R and the intersection Un by all integers n is equal to zero. ii) There exist two elements y ∈ U1 and y−1 ∈ U−1 such that yy−1 = y−1 y = 1; by considering the condition i), this implies that 1 = yy−1 ∈ U0 , i. e., U0 is a unitary subring in R. iii) For any family un ∈ Un , un+1 ∈ Un+1 , un+2 ∈ Un+2 , . . . , there exists an element u ∈ Un (the generalized infinite sum of elements ui ) such that, for all positive integers k, the inclusion n+k

(u − ∑ ui ) ∈ Un+k+1 i=n

holds. iv) The canonical ring homomorphism from the ring U0 onto its factor ring A = U0 /U1 splits, i. e., there exists an embedding π: A → R such that its composition with the canonical homomorphism U0 onto A is equal to the identity automorphism of the ring A. The coefficient ring of a Laurent ring If R is a ring which satisfies conditions i) and ii) of the second definition of a Laurent ring, then the factor ring A = U0 /U1 is called the coefficient ring of R. Below, we will prove that skew Laurent series rings and pseudo-differential operator rings are Laurent rings such that the notion of the coefficient ring in them coincides with the ordinary definition.

50 | 6 Equivalent definitions of Laurent rings

6.4 Remarks on the second definition of a Laurent ring The condition i) requires that R is a ℤ-filtered ring. The remaining conditions impose additional restrictions to this filtration. The condition ii) is specific for Laurent series with negative degrees of the variable. For example, the formal power series ring satisfies all the above conditions, except ii). The condition iii) is specific for infinite series. For example, the Laurent polynomial ring A[x, x−1 ] satisfies all the above conditions, except iii). Finally, the condition iv) is optional for the proof of some assertions; it requires the existence of a natural embedding of the coefficient ring in the generalized series ring. In contrast to the first definition 6.2, the second definition 6.3 of the Laurent ring is symmetrical with respect to multiplication on the right or left.

6.5 The equivalence of two definitions of a Laurent ring Any Laurent ring R in the sense of the first definition 6.2 also is a Laurent ring in the sense of the second definition 6.3 with family of subsets Un = π(Vn ) and the ring U0 /U1 is isomorphic to the ring A. In addition, every Laurent ring R in the sense of the second definition 6.3 also is a Laurent ring in the sense of the first definition 6.2 with coefficient ring A = U0 /U1 . ◁ Let R be a Laurent ring in the sense of the first definition 6.2 with coefficient ring A. We set Un = π(Vn ) for every integer n. The condition i) is directly verified; we can take y = π(x) and y−1 = π(x−1 ) as two mutually inverse elements for the condition ii). Since the ring A((x)) satisfies the relations A ∩ V1 = 0 and A + V1 = V0 , the relations π(A) ∩ U1 = 0 and π(A) + U1 = U0 hold in the ring R. Therefore, the ring U0 /U1 is isomorphic to the ring π(A); consequently, it is isomorphic to the ring A. In addition, the ring π(A) ≅ U0 /U1 is contained in the ring U0 ; therefore, the canonical homomorphism U0 → U0 /U1 splits, which proves the condition iv). It remains to prove that the condition iii) holds. Indeed, let us have a family of elements un ∈ Un , un+1 ∈ Un+1 , un+2 ∈ Un+2 , . . . of the ring R. Let fk = π −1 (uk ) ∈ Vk . We denote by fk,i the coefficient of the series fk such that the relation fk = fk,k xk + fk,k+1 x k+1 + ⋅ ⋅ ⋅ holds. We take a series f such that its coefficient of x k is equal to ∑ki=n fi,k for integers k which are not less than n (we assume that coefficients of degrees, which are less than

6.5 The equivalence of two definitions of a Laurent ring

| 51

n, are equal to zero). Now we set u = π(f ). It is directly verified that the element u satisfies the condition iii) as a “generalized infinite sum” of elements un . Now let R be a Laurent ring in the sense of the second definition 6.3 and A = U0 /U1 its coefficient ring. Let π be an embedding from the ring A in U0 which exists by iv). We extend the embedding π to an embedding from A((x)) in R, as follows. By the condition ii), we can set π(x) and π(x−1 ) equal to two mutually inverse elements such that π(x) ∈ U1 and π(x−1 ) ∈ U−1 . For any element a of the ring A and any positive integer n, we set n

π(axn ) = π(a)(π(x)) ∈ Un

n

and π(ax−n ) = π(a)(π(x −1 )) ∈ U−n .

Now for every series f = fn xn + fn+1 xn+1 + ⋅ ⋅ ⋅ , we set π(f ) to be equal a “generalized infinite sum” of elements π(fk xk ) for integers k which are not less than n. Such a “generalized infinite sum” exists by the condition iii). It is directly verified that the condition 1) of the first definition 6.2 holds, π is a well defined homomorphism of left modules, which carries out an embedding A((x)) in R, and π(Vn ) ⊆ Un for any integer n (where Vn is the set of all series which contain x with degrees not less that n). It is easy to see that the condition 4) of the first definition 6.2 holds, by the definition of the homomorphism π. Now we prove that π is a surjective mapping. Let r be an arbitrary element of the ring R. We denote by n the largest integer such that r is contained in Un (by the condition i), such an integer n exists). Then rπx −n is contained in U0 . Let rn be the image of the element rπx−n under the canonical homomorphism U0 → U0 /U1 = A. Then the element r − π(rn xn ) is contained in Un+1 . By applying the procedure applied to r to the element r − π(rn xn ), we find that the element r − π(rn xn ) − π(rn+1 x n+1 ) is contained in Un+2 . We continue this procedure to infinity and obtain a sequence of elements rk of A. It is directly verified that, for the series f = rn xn + rn+1 xn+1 + ⋅ ⋅ ⋅ , the relation π(f ) = r holds. Therefore, π(A((x))) = R; thus, π is an isomorphism of left modules over A. It also follows from the construction that π −1 (Un ) ⊆ Vn (by considering π(Vn ) ⊆ Un , we obtain π(Vn ) = Un ); therefore, the condition 3) of the first definition 6.2 holds. Indeed, π(Vn )π(Vm ) = Un Um ⊆ Un+m = π(Vn+m ). Therefore, it is proved that all conditions of the first definition 6.2 hold; the proof is completed. ▷

52 | 6 Equivalent definitions of Laurent rings Remark 6.6. Since the above two definitions are equivalent, we sometimes will unite these definitions in one definition and will say on conditions 1), 2), 3), 4), i), ii), iii), iv) of the definition of a Laurent ring. In addition, since there exists an embedding from the coefficient ring in the Laurent ring, we often assume that the coefficient ring is contained in the Laurent ring. We will show that the notion a “generalized infinite sum” defined in the second definition 6.3 of a Laurent ring is consistent with the formal infinite sum in the ring A((x)).

6.7 Consistency of generalized infinite sums with formal infinite sums in the ring A((x)) Let R be a Laurent ring with coefficient ring A and let π the bijection from A((x)) onto R from the first definition 6.2 of a Laurent ring. Then, for any sequence of elements an , an+1 , an+2 , . . . of the ring A, the elements π(ai x i ) form a “generalized infinite sum” which is equal to π(an xn + an+1 xn+1 + ⋅ ⋅ ⋅). ◁ Indeed, the element π(ai xi ) is contained in Ui = π(Vi ); therefore, the elements π(ai xi ) form a “generalized infinite sum”. It remains to prove that, for any k > n, the difference i≤k

π(an xn + an+1 xn+1 + ⋅ ⋅ ⋅) − ∑ π(ai x i ) i=n

is contained in Uk+1 . However, i≤k

π(an xn + an+1 xn+1 + ⋅ ⋅ ⋅) − ∑ π(ai x i ) i=n

i≤k

= π(an xn + an+1 xn+1 + ⋅ ⋅ ⋅) − π(∑ ai x i ) = π(ak+1 x which completes the proof. ▷

k+1

+ ak+2 x

k+2

i=n

+ ⋅ ⋅ ⋅) ∈ π(Vn ) = Un ,

7 Generalized Laurent rings Some results will be proved for a larger class of rings than the class of Laurent rings: namely, for the class of rings satisfying only conditions i)–iii) of the second of Definition 6.3 of a Laurent ring. In 9.11, we will give an example of the ring of fractional n-adic numbers Qn which satisfies conditions i)–iii), but does not satisfy the condition iv).

7.1 Generalized Laurent rings The rings, which satisfy only conditions i)–iii) of the second of Definition 6.3 of a Laurent ring, are called generalized Laurent rings. Namely, a ring R is called a generalized Laurent ring if R contains a family of additive subgroups {Ui | −∞ < i < +∞} which satisfy the properties i)–iii). i) For any two integers n and k, inclusions Un+1 ⊆ Un and Un Uk ⊆ Un+k hold; in particular, this implies that U0 is a subring in R and U1 is a two-sided ideal in U0 . In addition, the union Un by all integers n is equal to the ring R and the intersection Un by all integers n is equal to zero. ii) There exist a pair of elements y ∈ U1 and y−1 ∈ U−1 such that yy−1 = y−1 y = 1; by considering the condition i), this implies that 1 = yy−1 ∈ U0 , i. e., U0 is a unitary subring in R. iii) For any family un ∈ Un , un+1 ∈ Un+1 , un+2 ∈ Un+2 , . . . , there exists an element u ∈ Un (a generalized infinite sum of elements ui ) such that n+k

(u − ∑ ui ) ∈ Un+k+1 i=n

for all positive integers k.

7.2 Properties of generalized infinite sums Under the condition iii) of Definition 6.3, “generalized infinite sums” were defined for some families of summands. We will show below that this is simply the sum of a series which absolutely converging in some topology. Now we prove several useful simple properties of this sum. Let R be a generalized Laurent ring. In 7.2, we will denote a “generalized infinite sum” by an ordinary sign ∑. 1. A generalized sum of elements un ∈ Un , un+1 ∈ Un+1 , un+2 ∈ Un+2 , . . . https://doi.org/10.1515/9783110702248-007

54 | 7 Generalized Laurent rings is uniquely defined; i. e., there exists exactly one element which satisfies the condition iii) from Definition 6.3. 2. For any integer n, the relation ∑+∞ i=n 0 = 0 holds. 3. If the elements un ∈ Un , un+1 ∈ Un+1 , un+2 ∈ Un+2 , . . . and the elements vm ∈ Um , vm+1 ∈ Um+1 , vm+2 ∈ Um+2 , . . . form a “generalized infinite sum”, and there exists a bijective mapping η from nonnegative integers onto non-negative integers such that un+i = vm+η(i) for all nonnegative integers i, then the relation +∞

+∞

i=n

i=m

∑ ui = ∑ vi

holds. 4. Let r be an arbitrary element of the ring R and let the elements un ∈ Un , un+1 ∈ Un+1 , un+2 ∈ Un+2 , . . . form a “generalized infinite sum”. Then the relations +∞

+∞

+∞

+∞

i=n

i=n

i=n

i=n

r ∑ ui = ∑ rui ,

5.

∑ ui r = ∑ ui r,

hold and their right parts are always defined. If un ∈ Un , un+1 ∈ Un+1 , un+2 ∈ Un+2 , . . . are elements which form a “generalized infinite sum”, then, for any integer m > n, the relation +∞

m−1

+∞

i=n

i=n

i=m

∑ ui = ∑ ui + ∑ ui

holds, where the expression ∑m−1 i=n ui denotes an ordinary finite sum and the right part is always defined. 6. For any family of elements {ui,j | n ≤ i < +∞, m ≤ j < +∞} such that ui,j ∈ Ui+j , the relation +∞ +∞

+∞ i−n

i=n j=m

i=n+m j=m

∑ ∑ ui,j = ∑ ∑ ui−j,j

holds, and both of its parts are always defined.

7.2 Properties of generalized infinite sums | 55

7.

For any family of elements {ui,j | n ≤ i < +∞, m ≤ j < +∞} such that ui,j ∈ Ui+j , the relation +∞ +∞

+∞ +∞

i=n j=m

j=m i=n

∑ ∑ ui,j = ∑ ∑ ui,j

holds and both of its parts are always defined. ◁ 1.

Indeed, we assume that there exist two distinct elements u and v which satisfy the condition iii). Then, by assumption, for every positive integer k, inclusions k

u − ∑ ui ∈ Uk+1 i=n

k

and v − ∑ ui ∈ Uk+1 i=n

hold. Therefore, u − v ∈ Uk+1 . Therefore, since k is arbitrary, we have u − v = 0, as required. 2. The assertion is directly verified. 3. We denote by u and v the infinite sum of ui and the infinite sum of vi , respectively. To prove that u = v, it is sufficient to prove that, for all integers k exceeding some integer, the inclusion u − v ∈ Uk holds. Let k 󸀠 be the maximal value taken by the function m + η(i) for i ≤ k − n and let k 󸀠󸀠 be the maximum of two numbers k and k 󸀠 . Then k

k

k 󸀠󸀠

k 󸀠󸀠

i=n

i=n

i=m

i=m

u − v = (u − ∑ ui ) + (∑ ui − ∑ vi ) − (v − ∑ vi ), where the first and the third summand are contained in Uk+1 by the definition “generalized infinite sum”. It remains to prove that the second summand is contained in Uk as well. 󸀠󸀠 Indeed, it follows from the assumption and the choice of k 󸀠󸀠 that the sum of ∑ki=m vi contains all the same terms as the sum of ∑ki=n ui and also some additional terms vi , for which the relation η−1 (i − m) > k − n holds. However, if η−1 (i − m) > k − n, then vi = un+η−1 (i−m) ∈ Uk+1 , which completes the proof. 4. Let the lowest degree of the element r be equal to m. If ui is contained in Ui , then ui r, rui are contained in Ui+m ; therefore, the right parts of the relations that we prove are defined. The proofs of these relations are similar; for definiteness, we prove the first relation. We have to prove that the element r ∑+∞ i=n ui satisfies the condition iii) of Definition 6.3 for a family of elements {rui }. Indeed, for all integers k, we have +∞

k

+∞

k

i=n

i=n

i=n

i=n

r ∑ ui − ∑ rui = r( ∑ ui − ∑ ui ) ∈ Un+m , as required.

56 | 7 Generalized Laurent rings 5.

Indeed, if a family of elements un , un+1 , . . . satisfy the conditions of a “generalized infinite sum”, then the family um , um+1 , . . . also satisfy them. For any integer k > m, we have +∞

m−1

k

+∞

k

i=n

i=n

i=m

i=n

i=n

( ∑ ui − ∑ ui ) − ∑ ui ∑ ui − ∑ ui ∈ Uk+1 , which means, corresponding to the condition iii) of Definition 6.3, that +∞

m−1

+∞

i=n

i=n

i=m

∑ ui − ∑ ui = ∑ ui ,

as required. 6. Indeed, it follows from the condition iii) of Definition 6.3 that, for all i ≥ n, we have +∞

∑ ui,j ∈ Ui+m .

j=m

Therefore, the left part of the required relation is always defined. Since i−n

∑ ui−j,j ∈ Ui

j=m

the right part is defined as well; it remains to prove that the two parts are equal. Indeed, it is sufficient to prove that, for all sufficiently large integers k, the difference +∞ +∞

+∞ i−n

i=n j=m

i=n+m j=m

∑ ∑ ui,j − ∑ ∑ ui−j,j

is contained in Uk+1 . For this purpose, we will show that the difference k−m +∞

k

i−n

∑ ∑ ui,j − ∑ ∑ ui−j,j

i=n j=m

i=n+m j=m

is contained in Uk+1 . It is sufficient to prove that the difference k−m k−n

k

i−n

∑ ∑ ui,j − ∑ ∑ ui−j,j

i=n j=m

7.

i=n+m j=m

is contained in Uk+1 . However, the last relation contains only finite sums and directly follows from the property that ui,j ∈ Ui+j . The required relation follows from Item 6 applied to both the right part and the left part of the required relation. ▷

7.3 Remark on the symbol ∑ in 7.2

| 57

7.3 Remark on the symbol ∑ in 7.2 In 7.2, the symbol ∑ denotes a “generalized infinite sum”. We use this designation for generalized sums in 7.2 only to avoid a contradiction with designation of formal power series in Laurent series rings and pseudo-differential operator rings; however, in these rings, a formal infinite power series is a “generalized infinite sum” of its terms.

7.4 Some notation and definitions for generalized Laurent rings and properties of such rings Let R be a generalized Laurent ring and {Ui } a family of its subsets as in the second of Definition 6.3 of a Laurent ring. For every element of U0 , its image under the canonical homomorphism from U0 onto A = U0 /U1 is called its constant term. In the case of the Laurent series ring, this definition coincides with a natural definition of a constant term; therefore, the constant term (if it is defined) of the element f is denoted by f0 . a. It is easy to see that the constant term of the sum (resp., the product) of two elements from U0 is equal to the sum (resp., the product) of their constant terms. For every non-zero element f from R, we call by its lowest degree an integer n such that f is contained in Un and it is not contained in Un+1 (for the ordinary Laurent series ring, the lowest degree coincides with the degree of the lowest term). Sometimes, it will be convenient to assume that the lowest degree of zero is equal to plus infinity. The lowest degree is uniquely defined. The elements with lowest degree 0 coincide with all elements of U0 with non-zero constant term. b. Every non-zero element f of the ring R can be represented as the product of the form uyn , where u is an element of U0 with non-zero constant term, y is an invertible element described in condition ii) from the second of Definition 6.3 of a Laurent ring, and n is the lowest degree of f . It follows from this that, for any integers n and m, the relation yn Um = Un+m holds. c. Let R be a generalized Laurent ring and y some invertible element described in the condition ii) of Definition 6.3. If u0 + U1 is some element of the coefficient ring U0 /U1 , then, for any integer n, the element yn u0 y−n + U1 also is an element of the coefficient ring which does not depend on the choice of a specific representative u0 (since yn U1 y−n = U1 ). Therefore, the mapping φ: u0 + U1 → yu0 y−1 + U1 defines an automorphism of the coefficient ring and φn takes the element u0 + U1 to yn u0 y−n + U1 . In the case of the skew Laurent series ring A((x, φ)) and the element y, which is equal to x, this automorphism coincides with the automorphism defining a skew multiplication in the skew Laurent series ring. Therefore, we use the same designation, φ, for it; it is called the twisting automorphism of a generalized Laurent ring. Generally speaking, the twisting automorphism depends on the choice of the specific element y. In the case of the pseudo-differential operator

58 | 7 Generalized Laurent rings ring and the element y = t −1 , this automorphism is the identity automorphism of the coefficient ring. For every subset P of the generalized Laurent ring R, we denote by λ(P) the image of the set U0 ∩ P under the canonical homomorphism from U0 onto U0 /U1 (i. e., λ(P) is the set of all constant terms of all series from U0 ∩ P). d. It is directly verified that, if P is a left (right) ideal of the ring R, then λ(P) is a left (right) ideal of the coefficient ring A = U0 /U1 . Therefore, the mapping λ carries out mapping from the lattice of left (right) ideals of the ring R in the lattice of left (right) ideals of the ring A and this mapping preserves the inclusion relation. In addition, if P is a non-zero right (left) ideal of the ring R, then λ(P) is a non-zero right (left) ideal (indeed, if f ∈ P and n is an integer such that f ∈ Un \Un+1 , then fx −n ∈ U0 ∩ P or x−n f ∈ U0 ∩ P; in addition, the elements fx −n and x −n f have nonzero constant terms). Now we prove some auxiliary assertion on generalized Laurent rings. Lemma 7.5. Let R be a generalized Laurent ring with coefficient ring A and A is a domain. Then, for any two non-zero elements f , g of R, the lowest degree of their product is equal to the sum of their lowest degrees. ◁ Indeed, if n is the lowest degree of the element f and k is the lowest degree of the element g, then there exist two elements f 󸀠 and g 󸀠 from U0 \U1 such that f = yn f 󸀠 and g = g 󸀠 yk . Since the elements f 󸀠 and g 󸀠 have non-zero constant terms, their product f 󸀠 g 󸀠 has the non-zero constant term as well; consequently, it is contained in U0 \U1 . The product fg = yn f 󸀠 g 󸀠 yk is contained in Un+k . If is contained also in Un+k+1 , then the product f 󸀠 g 󸀠 = y−n fgy−k is contained in U1 , which is impossible. Therefore, the product fg = yn f 󸀠 g 󸀠 yk is contained in Un+k \Un+k+1 , which one is required to prove. ▷ Lemma 7.6. Let R be a generalized Laurent ring with coefficient ring A and P a right ideal in R. We assume that there exist elements f1 , f2 , . . . , fn of P ∩ U0 such that, for every element g from P ∩ U0 , the inclusion g0 ∈ f1,0 A + f2,0 A + ⋅ ⋅ ⋅ + fn,0 A holds, where fi,0 is the constant term of the element fi and g0 is the constant term of the element g. Then the right ideal P is generated by n elements f1 , f2 , . . . , fn . ◁ We denote by Q the right ideal f1 R + f2 R + ⋅ ⋅ ⋅ + fn R of the ring R. Since all elements fi are contained in the ideal P and Q = f1 R+f2 R+⋅ ⋅ ⋅+fn R, we have Q ⊆ P. We assume that the assertion of the lemma is not true. Then there exists a series h ∈ P such that h ∈ ̸ Q. Without loss of generality, we can assume that h ∈ U0 (otherwise, we can multiply h by the element y from the condition ii) of Definition 6.3 in the corresponding degree).

7.8 Remarks | 59

Let h0 be the constant term of the element h. By assumption, h0 = f1,0 a1,0 + f2,0 a2,0 + ⋅ ⋅ ⋅ + fn,0 an,0 for some elements a1,0 , . . . , an,0 of the ring A. We consider the element (h − (f1 a1,0 + ⋅ ⋅ ⋅ + fn an,0 ))y−1 = h󸀠 ∈ U0 . The element h󸀠 is contained in P ∩ U0 and we apply the assumption of the lemma to it: h󸀠0 = f1,0 a1,1 + f2,0 a2,1 + ⋅ ⋅ ⋅ + fn,0 an,1 . We take the series h󸀠󸀠 to be equal to (h󸀠 − (f1 a1,1 + ⋅ ⋅ ⋅ + fn an,1 ))y−1 . The sequence h, h󸀠 , h󸀠󸀠 , . . . can be continued to infinity. It is directly verified that we have the relation (in the sense of “generalized infinite sums” which exist by the condition iii) of Definition 6.3) h = f1 (a1,0 + a1,1 y + a1,2 y2 + ⋅ ⋅ ⋅) + ⋅ ⋅ ⋅ + fn (an,0 + an,1 y + ⋅ ⋅ ⋅). This contradicts the assumption h ∈ ̸ Q. ▷ Proposition 7.7. Let R be a generalized Laurent ring with coefficient ring A. Then, if the constant term of the element r ∈ U0 is right (left) invertible in the ring A, then the element r is right (left) invertible in the ring R. In addition, if A is a division ring, then R is a division ring as well. ◁ Indeed, let the constant term of the element r be right invertible. Then we apply Lemma 7.6, where we set n = 1, f1 = r and P = R. We obtain R = P = rR, i. e., the element r is right invertible. Now let A be a division ring, then it follows from the already proved results that every element u of U0 with non-zero constant term is right (and left) invertible. However, every non-zero element r of the ring R can be represented in the form of the product uyn , where u ∈ U0 has the non-zero constant term and y is an invertible element. Therefore, r is the product of two invertible elements; consequently, it is invertible. ▷

7.8 Remarks 1. The analogue of Lemma 7.6 for left ideals is true as well. 2. In connection to Proposition 7.7, we note that the following assertion easily follows from Lemma 7.6 (its partial cases for skew Laurent series rings and for pseudodifferential operator rings are well known):

60 | 7 Generalized Laurent rings A Laurent ring is a division ring if and only if its coefficient ring is a division ring (see 10.1). For generalized Laurent rings, this assertion is not true; the corresponding example will be constructed below (the ring of fractional pn -adic numbers Qpn , n > 1, which is a division ring, while its coefficient ring is not a domain). The notion a “generalized infinite sum”, defined in the second of Definition 6.3 of a Laurent ring, can be considered from a topological point of view, by considering it as the sum of the series which absolutely converges in a certain topology.

7.9 Normed rings We recall that a ring A with function ‖⋅‖ in A → [0; +∞) defined on A is called a normed ring if the following conditions hold. 1) The relation ‖x‖ = 0 holds if and only if x = 0. 2) For any two elements x and y of the ring A, the inequality ‖x + y‖ ≤ ‖x‖ + ‖y‖ holds. 3) For any two elements x and y of the ring A, the inequality ‖xy‖ ≤ ‖x‖ ⋅ ‖y‖ holds. 4) The relations ‖1‖ = ‖ − 1‖ = 1 hold. In a normed ring, as usual, the metrics ρ(x, y) = ‖x − y‖ and the related topology are defined. In addition, the ring operations of multiplication and addition are uniformly continuous functions in two variables. We will show that on the generalized Laurent ring we can naturally define a topology in agreement with a norm. Theorem 7.10. Let R be a generalized Laurent ring and f : ℤ → (0; +∞) a properly monotonously decreasing function in integral argument with the following properties: f (n + m) ≤ f (n)f (m),

f (0) = 1,

lim f (n) = 0;

n→+∞

for example, we can take the function f (n) = 2−n . Then we can define a norm on R with the properties 1)–4). 1) ‖0‖ = 0 and for a non-zero element r from R with lowest degree n, its norm ‖r‖ is equal to f (n). 2) With such a norm, R is a normed ring and the topology, generated by the norm, does not depend on the choice of the specific function f . 3) R is a complete metrical space and every unitary subring R󸀠 in R which satisfies the relation λ(R󸀠 ) = A and contains at least one pair of mutually inverse elements from U1 and U−1 is everywhere dense in R. 4) The mapping π from A((x)) in R, as in the definition of a Laurent ring, is a homeomorphism of topological spaces if we can define a topology in A((x)) in the same way as in R.

7.9 Normed rings | 61

◁ Fulfillment of all conditions of the definition of a normed ring for the ring R with the specified norm follows from properties of the function f and from the property that the lowest degree of the product of two elements of the ring R is not less than the sum of their lowest degrees and the lowest degree of the sum of two elements of the ring R is not less than the minimum of their lowest degrees. To verify that the topology, generated by the norm, does not depend on the choice of the function f , it is sufficient to prove that the fundamental system of neighborhoods of zero consisting of neighborhoods of the form Uε󸀠 = {r | ‖r‖ < ε} for 0 < ε < 1 does not depend on the choice of the function f . Indeed, Uε󸀠 = {r | ‖r‖ < ε} = ⋃ Un = Ug(ε) , f (n) 0 and only of them. Therefore, this system does not depend on the choice of the function f . We verify that R is a complete metrical space. Indeed, let us have a Cauchy sequence {rn } of elements of the ring R. We consider a subsequence {rn󸀠 } such that, for 󸀠 all positive integers n, the inequality ‖rn+1 − rn󸀠 ‖ ≤ f (n) holds. Then, for all positive 󸀠 integers n, the inclusion rn+1 − rn󸀠 ∈ Un holds. By the condition iii) of the second of Definition 6.3 of a Laurent ring, this means that there exists a “generalized infinite 󸀠 sum” of s elements of all rn+1 − rn󸀠 for n ≥ 1. Then the element r = r1 + s is the limit of 󸀠 the sequence {rn }; therefore, it is the limit of the sequence {rn } as well. We prove this assertion. Indeed, by the definition of a “generalized infinite sum” for all positive integers n, we have the inclusion n−1

󸀠 r − rn󸀠 = s − ∑ (ri+1 − ri󸀠 ) ∈ Un , i=1

which means that ‖r − rn󸀠 ‖ ≤ f (n). Therefore, it is easy to see that the element r is equal to the limit of the sequence {rn󸀠 }. Now let R󸀠 be a subring in the ring R which satisfies the relation λ(R󸀠 ) = A and contains two mutually inverse elements y from U1 and y−1 from U−1 . Let r be an arbitrary non-zero element of the ring R with lowest degree n. We have to prove that, for any integer k, there exists an element rk ∈ R󸀠 such that r − rk ∈ Uk . We will use induction on k − n. Indeed, if n − k ≤ 0, then we can take rk = 0. Now let k − n be an arbitrary integer. Then we consider the constant term rn of the element ry−n ∈ U0 . By assumption, λ(R󸀠 ) = A; therefore, rn is contained in λ(R󸀠 ); therefore, R󸀠 ∩ U0 contains an element r 󸀠 with constant term rn . Then the difference ry−n −r 󸀠 is contained in U1 ; therefore, the difference r−r 󸀠 yn is contained in Un+1 . Then we apply the induction hypothesis to the difference r − r 󸀠 yn ; thus, there exists an element

62 | 7 Generalized Laurent rings r 󸀠󸀠 of R󸀠 such that r −r 󸀠 yn −r 󸀠󸀠 is contained in Uk . Since the element r 󸀠 yn +r 󸀠󸀠 is contained in R󸀠 , the proof is completed. It remains to prove that π is a homeomorphism. Indeed, π is a bijection; in addition, π and π −1 preserve fundamental systems neighborhoods of any point: π −1 (r + Un ) = π −1 (r) + Vn

and π(g + Vn ) = π(g) + Un ;

therefore, π is a homeomorphism. ▷ Remark 7.11. By considering the norm and the topology defined in this proposition, “a generalized infinite sum” in the definition of a Laurent ring is simply the sum of an absolutely converging series. Completeness of a Laurent ring as a metrical space is directly related to condition iii) of the second of Definition 6.3. So, if a Laurent polynomial ring A[x, x−1 ] does not satisfy condition iii), we can define a norm and topology in A[x, x−1 ] similarly, but the obtained space will not be complete. In addition, the ring A[x, x−1 ] satisfies the conditions required in the proposition, since it is a subring in A((x)); therefore, A[x, x−1 ] is an everywhere dense subset of the space A((x)).

8 Properties of Laurent rings 8.1 Coefficients and constant terms of Laurent rings Let R be a Laurent ring with coefficient ring A and let π be the fixed mapping from A((x)) into R, as in the first of Definition 6.2 of a Laurent ring. Then, if f = ∑ fi x i is a series from the Laurent series ring A((x)), then we call elements fi from the ring A left coefficients of the element π(f ) ∈ R. The mapping π is the bijection from A((x)) onto R such that, for every element of R, there exists one and only one family of left coefficients (under the fixed mapping π). It is directly verified that if the element u is contained in U0 , then its constant term coincides with its left coefficient u0 , regardless of the selected mapping π from A((x)) into R.

8.2 The mapping μ: 2A → 2R for a Laurent ring R Let R be a Laurent ring with coefficient ring A. For every subset B of the ring A, we denote by μ(B) the set of all those elements of the ring R whose left coefficients are contained in B. The following properties are directly verified. 1. If B is a right ideal of the ring A, then μ(B) is a right ideal of the ring R and λ(μ(B)) = B. 2. If B is a right ideal of the ring A, then the right ideal μ(B) of the ring R is closed with respect to “generalized infinite sums” (as in the condition iii) of the second of Definition 6.3 of a Laurent ring). 3. The mapping μ carries out an embedding of the lattice of right ideals of the ring A in the lattice of right ideals of the ring R (this embedding is a lattice homomorphism preserving infinite sums and intersections). 4. For any principal right ideal aA of the coefficient ring A, we have the relation μ(aA) = π(a)R. Remark 8.3. In contrast to the mapping λ, the mapping μ is not defined symmetrically with respect to right or left multiplication. It is possible to define it for right coefficients and then μ would induce an embedding from the lattice of left ideals. In addition, the existence of such a mapping implies the condition iv) of the second of Definition 6.3 and the mapping itself depends on the choice of a specific bijective mapping π: A((x)) → R, since it uses the notion of left coefficients. Lemma 8.4. Let R be a Laurent ring with coefficient ring A. If B is a maximal right ideal of the ring A, then μ(B) is a maximal right ideal of the ring R. ◁ Indeed, let r be an element of the ring R not contained in μ(B). Then it is sufficient to prove that rR + μ(B) = R. https://doi.org/10.1515/9783110702248-008

64 | 8 Properties of Laurent rings Since the element r is not contained in μ(B), some of its left coefficients are not contained in B, we choose from them the coefficient with the smallest number (which stands at the lowest degree of variable in A((x))), let this coefficient be rn . Then, subtracting elements contained in μ(B) from the element r, we can ensure that all left coefficients r with numbers less than n are equal to zero. Therefore, we assume that rn has the least number among non-zero coefficients. Moving from the element r to the element rπ(x−n ), we can be sure that the number n is equal to zero. Then the element r is contained in U0 and its constant term r0 is not contained in B. Since B is a maximal right ideal of the ring A, there exist elements a of A and b of B with r0 a + b = 1. Then the constant term of the element rπ(a) + π(b) is equal to 1 (and this element is contained in rR + μ(B)); consequently, by Proposition 7.7, it is invertible in the ring R. Therefore, rR + μ(B) = R, as required. ▷ Proposition 8.5. Let R be a Laurent ring with coefficient ring A. Then the Jacobson radical J(R) of the ring R is contained in μ(J(A)), where J(A) is the Jacobson radical of the ring A. ◁ Let {Bi }i∈I be the set of all maximal right ideals of the ring A. Since the radical J(A) coincides with the intersection of all right ideals Bi , the right ideal μ(J(A)) coincides with the intersection of right ideals μ(Bi ) corresponding with them. By Lemma 8.4, all right ideals μ(Bi ) are maximal right ideals of the ring R; therefore, each of them contains the radical J(R). Therefore, we find that J(R) is contained in μ(J(A)). ▷ Lemma 8.6. Let R be a Laurent ring with coefficient ring A and let P be a two-sided ideal of the ring A such that μ(P) is a two-sided ideal of the ring R. Then we can naturally define the structure of a Laurent ring on the factor ring R/μ(P) such that the coefficient ring is isomorphic to A/P. ◁ Indeed, we verify fulfillment of the first of Definition 6.2 of a Laurent ring. Let B = A/P. We construct a mapping χ from B((x)) into R/μ(P). Let b be a series in B((x)) and let {bi } be its coefficients with b = ∑ bi xi . For every i, let us have the relation bi = ai + P, where {ai } is some family of elements of the ring A. Then we denote by a the series ∑ ai xi and we set χ(b) = π(a) + μ(P). It is directly verified that this definition does not depend on the choice of ai and that all properties, required in the first of Definition 6.2 of a Laurent ring, hold. ▷ To construct specific examples of Laurent rings, we have to define some multiplication rule for series in A((x)) (since the addition rule is fixed in the definition). However, direct verification as regards axioms of the ring (associativity law for multiplication, especially) is a very difficult procedure. Therefore, it will be useful to prove several general lemmas which allow one to simplify verification axioms of the ring and construction multiplication in the ring. Lemma 8.7. Let A be a ring and f a mapping to which every monomial of the form ax m , where a is an element of the ring A and m is an arbitrary integer, matches the series from

8.2 The mapping μ: 2A → 2R for a Laurent ring R

|

65

the Laurent series ring A((x)) and for all a, b in A and all integers m, we have the relation f ((a + b)xm ) = f (ax m ) + f (bxm ). Let there exist an integer n such that, for any element a in A and for any integer m, the lowest degree of the series f (axm ) is not less than n + m. Then the mapping f can be uniquely extended to an endomorphism of f 󸀠 of the Abelian group A+ ((x)); so, the restriction f 󸀠 to the set of monomials of the form ax m coincides with f ; in addition, for any series r, the lowest degree of the series f 󸀠 (r) will not be less than n + m, where m is the lowest degree of the series r. In addition, if the relation f (caxn ) = cf (axn ) holds for some c ∈ A and all monomials n ax or the condition f (axn xj ) = f (ax n )xj holds for some integer j and all monomials ax n , then the same condition holds for the mapping f 󸀠 as well. If, for any element a of A and for any integer m, the lowest degree of the series f (axm ) is equal to n + m, then, for any series r, the lowest degree of the series f 󸀠 (r) is equal to n + m, where m is the lowest degree of the series r. ◁ For every fixed m, the mapping f defines a homomorphism from the additive Abelian group Ax m consisting of all monomials of the form axm into the Abelian group A+ ((x)). Therefore, the mapping f can be extended to a homomorphism from the direct sum of Abelian groups Axm into the Abelian group A+ ((x)). The direct sum of Abelian groups Axm over all integers m coincides with the Laurent polynomial ring A[x, x −1 ], so we can assume that f is a homomorphism from the Abelian group A+ [x, x −1 ] into the Abelian group A+ ((x)). The condition f (cax n ) = cf (axn ) or f (axn xj ) = f (axn )x j , if it was satisfied, meanwhile obviously remains. It is easy to see that if, for every element a of A and for any integer m, the lowest degree of the series f (ax m ) is n + m, then, for every Laurent polynomial r with lowest degree m, the lowest degree of the series f (r) is equal to n + m. i We prove that such an extension f 󸀠 exists. Let r be an arbitrary series ∑+∞ i=m fi x . For every integer k, we denote by r (k) the Laurent polynomial ∑ki=m fi x i . For k < m, we assume that r (k) = 0. We construct the series f 󸀠 (r). We see that the coefficient of x k of the series f 󸀠 (r) is equal to the coefficient of xk of the series f (r (k−n) ). It is easy to see that all coefficients of the series f 󸀠 (r) of powers of x, which are lower than n + m, are equal to zero; therefore, the series f 󸀠 (r) is well defined and its lowest degree is not less than n + m. It is obvious that the obtained mapping f 󸀠 is an endomorphism of the Abelian group A+ ((x)), since (r + s)(k) = r (k) + s(k) . It remains to prove that f (r) = f 󸀠 (r) for any Laurent polynomial r. Indeed, the coefficient of xk of the series f 󸀠 (r) is equal to a coefficient of the series (k−n) f (r ) of xk . Consequently, we have to prove that, for any integer k, the coefficient of xk of the series f (r (k−n) ) is equal to the coefficient of x k of the series f (r). For this purpose, it is sufficient to prove the inclusion f (r) − f (r (k−n) ) ∈ Vk+1 , where Vm denotes the set of all series whose lowest degrees are not less that m. However, f (r) − f (r (k−n) ) = f (r − r (k−n) ) and the polynomial r − r (k−n) is a sum of monomials with degree above k − n. Therefore, the series f (r − r (k−n) ) is the sum of series with degree exceeding k; consequently, it is contained in Vk+1 , as required.

66 | 8 Properties of Laurent rings By construction, it is easy to verify that if f (cax n ) = cf (axn ) or f (axn x j ) = f (ax n )x j , then the condition is preserved for the mapping f 󸀠 and that if for any Laurent polynomial r with lowest degree m, the lowest degree of the series f (r) is equal to n + m, then, for any series r with lowest degree m, the lowest degree of the series f 󸀠 (r) is equal to n + m. Now we prove that such an extension f 󸀠 is unique. Indeed, if there are two such extensions, then their difference g also is an endomorphism of the Abelian group A+ ((x)) and g(A[x, x−1 ]) = 0. In addition, for any series r, the lowest degree of the series g(r) is not less than n + m, where m is the lowest degree of the series r. We assume that there exists a series r such that g(r) is non-zero. Let k be the lowest degree of the series g(r). Then the series g(r − r (k−n) ) = g(r) − g(r (k−n) ) = g(r) has the lowest degree k. On the other hand, since the lowest degree of the series r − r (k−n) is not less than k − n + 1, the lowest degree of the series g(r − r (k−n) ) is not less than k + 1. This is a contradiction. ▷ Remark 8.8. From the point of view of the topology defined in Theorem 7.10, Lemma 8.7 is a corollary concerning the property that a uniformly continuous mapping defined on the everywhere dense subset A[x, x −1 ] can be extended to the whole complete metrical space and is unique. Sometimes, it is convenient not to use Lemma 8.7 but its simple corollary. Proposition 8.9. Let A be a ring, n be an integer, and let f : A+ → Vn be a homomorphism of the Abelian groups which matches to every element a ∈ A a series from the Laurent series ring A((x)) such that its lowest degree is not less that n. Then the mapping f can be uniquely extended to an endomorphism of f 󸀠 of the Abelian group A+ ((x)) such that the restriction f 󸀠 to A coincides with f , and for all series r and all integers k, we have the relation f 󸀠 (rxk ) = f 󸀠 (r)xk ; in addition, for any series r, the lowest degree of the series f 󸀠 (r) is not less than n + m, where m is the lowest degree of the series r. If, for any element a of A, the lowest degree of the series f (a) is equal to n, then, for any series r, the lowest degree of the series f 󸀠 (r) will be equal to n + m, where m is the lowest degree of the series r. ◁ By considering the relation f 󸀠 (axm ) = f 󸀠 (a)xm , there exists a unique extension f 󸀠 of the mapping f from the Abelian group A+ onto the set of all monomials of the form ax m , where a is an element of the ring A and m is an arbitrary integer. To the mapping f 󸀠 , we can apply Lemma 8.7, which completes the proof. ▷

9 Laurent rings: examples, relation Lemma 9.1. Let A be a ring, let A((x)) be the Laurent series ring over A, and let ω(⋅, ⋅) be a function which to every pair of series from A((x)) matches the series from A((x)) and satisfies the following conditions 1)–7), where f , g, h in the relations denote arbitrary series from A((x)), n, m are arbitrary integers, and a and b are arbitrary elements of A. 1) ω(f + g, h) ≡ ω(f , h) + ω(g, h) and ω(f , g + h) ≡ ω(f , g) + ω(f , h). 2) The lowest degree of the series ω(f , g) is not less than the sum of the lowest degrees of the series f and g, while the lowest degree of the series ω(x, f ) is always exactly one greater than the lower degree of the series f , and the lowest degree of the series ω(x −1 , f ) is exactly one less than the lower degree of f . 3) ω(1, f ) ≡ f , ω(x, 1) = x and ω(x −1 , 1) = x−1 . 4) ω(af , gxn ) ≡ aω(f , g)xn . 5) ω(x n , g) ≡ ω(x, ω(xn−1 , g)) for n > 0 and ω(xn , g) ≡ ω(x −1 , ω(xn+1 , g)) for n < 0. 6) ω(x, ω(x −1 , a)) ≡ a. 7) ω(x −1 , ab) ≡ ω(ω(x−1 , a), b). Then we can define multiplication f ∘ g = ω(f , g) on the Abelian group A+ ((x)) such that A+ ((x)) satisfies all ring axioms. ◁ In the proof, f , g, h from the relations denote arbitrary series from A((x)), n and m are arbitrary integers and a, b are arbitrary elements of A. The proof is based on the following trick: if some relation β(f ) ≡ γ(f ) holds for all monomials f of the form axn and the functions β and γ satisfy the conditions of Lemma 8.7, then this relation holds for all series f from A((x)). This follows from the fact that, by Lemma 8.7, an extension of the function β − γ to the whole ring A((x)) is unique and identically equal to zero (it is directly verified that, if the functions β and γ satisfy conditions Lemma 8.7 for some integers nβ and nγ , then the function β − γ satisfies the conditions of Lemma 8.7 for min(nβ , nγ )). As a rule, the functions ω will be taken as functions β and γ with a fixed argument or its composition with itself. Therefore, by virtue of the conditions 1) and 2), the conditions of Lemma 8.7 will be satisfied. The main difficulty is the proof of associativity of multiplication defined by the function ω. We will prove the required relations. It follows from conditions 3) and 5) that ω(xn , 1) ≡ xn . Therefore, it follows from the condition 4) that ω(axn , 1) ≡ ax n for all a in A and, by Lemma 8.7, ω(f , 1) ≡ f for all f in A((x)). Therefore, by the condition 4) ω(f , xn ) ≡ fx n for all integers n, in particular ω(xn , xm ) ≡ xn+m . It follows from conditions 3) and 4) that ω(a, f ) ≡ af for all a and f . Linear endomorphisms β and β−1 of the Abelian group A+ ((x)), defined by the relations β(f ) = ω(x, f ) https://doi.org/10.1515/9783110702248-009

and β−1 (f ) = ω(x−1 , f ),

68 | 9 Laurent rings: examples, relation are injective, by the condition 2). It follows from conditions 6) and 4) that ω(x, ω(x −1 , axn )) ≡ ax n . Therefore, by Lemma 8.7, we obtain the relation ω(x, ω(x −1 , f )) ≡ f . Therefore, ββ−1 = 1A+ ((x)) . Thus, the endomorphism β is surjective. Since β is injective, it is an automorphism. Then β−1 coincides with the automorphism β−1 . It follows from conditions 3) and 5) that βn (f ) = ω(x n , f ) for all integers n. Therefore, for all integers n and m, we obtain ω(xn , ω(xm , f )) ≡ ω(xn+m , f ) ≡ ω(ω(xn , x m ), f ). It follows from these relations with the use of 4) that, for all integers n and m, for all a ∈ A and for all series g, we have the relation ω(axn , ω(xm , g)) ≡ ω(ω(axn , x m ), g). By Lemma 8.7, we obtain ω(f , ω(xm , g)) = ω(ω(f , x m ), g) for all series f and g and all integers m. It follows from conditions 7), 4) and the relations proved above that ω(x−1 , ω(a, bxn )) ≡ ω(x−1 , abxn ) ≡ ω(x−1 , ab)xn ≡ ω(ω(x−1 , a), b)xn ≡ ω(ω(x−1 , a), bxn ). By Lemma 8.7, we obtain the relation ω(x−1 , ω(a, f )) ≡ ω(ω(x−1 , a), f ). We substitute f = ω(xn , g) in the last relation and use the above relations. Thus, we obtain ω(x−1 , ω(axn , g)) ≡ ω(x−1 , ω(ω(a, xn ), g)) ≡ ω(x−1 , ω(a, ω(xn , g))) ≡ ω(ω(x−1 , a), ω(xn , g)) ≡ ω(ω(ω(x−1 , a), xn ), g) ≡ ω(ω(x−1 , axn ), g). Therefore, with the use of Lemma 8.7, we obtain the relation ω(x−1 , ω(f , g)) ≡ ω(ω(x−1 , f ), g). We use induction on n to prove the relation ω(x−n , ω(f , g)) ≡ ω(ω(x −n , f ), g) for all positive integers n. For n = 1, it is proved. Let us assume that it is proved for some n = k. By the condition 5), we can use the relation, proved for n = k and n = 1, to obtain ω(x−k−1 , ω(f , g)) ≡ ω(x−1 , ω(x−k , ω(f , g))) ≡ ω(x−1 , ω(ω(x−k , f ), g)) ≡ ω(ω(x−1 , ω(x−k , f )), g) ≡ ω(ω(x−k−1 , f ), g).

9 Laurent rings: examples, relation |

69

Therefore, the relation ω(x−n , ω(f , g)) ≡ ω(ω(x−n , f ), g) has been proved for all positive integers n (for n = 0, it follows from the condition 3)). Now let n > 0. Then we use the above to obtain ω(ω(xn , f ), g) ≡ βn (β−n (ω(ω(x n , f ), g))) ≡ ω(x n , ω(x−n , ω(ω(xn , f ), g)))

≡ ω(xn , ω(ω(x−n , ω(xn , f )), g)) ≡ ω(xn , ω(ω(x−n , ω(xn , f )), g)) ≡ ω(xn , ω(β−n (βn (f )), g)) ≡ ω(xn , ω(f , g)).

Therefore, the relation ω(xn , ω(f , g)) ≡ ω(ω(xn , f ), g) is proved for all integers n. By considering the condition 4), the relation ω(axn , ω(f , g)) ≡ ω(ω(axn , f ), g) has been proved. With the use of Lemma 8.7, this relation can be extended to all series. Thus, we obtain the relation ω(h, ω(f , g)) ≡ ω(ω(h, f ), g). Therefore, the associativity is proved. The distributivity of multiplication, defined by the function ω, directly follows from the condition 1), the relation ω(f , 1) ≡ f has been proved above, and the relation ω(1, f ) ≡ f holds by condition 3). Therefore, the Abelian group A+ ((x)) with multiplication ω is a ring. ▷ Lemma 9.2. Let A be a ring with automorphism φ and Δ: A+ → A+ [[x]] an arbitrary homomorphism of Abelian groups which matches to every element of the ring A the series without negative degrees of the variable with coefficients from A. Then there exists a unique function ω(⋅, ⋅), which matches to every pair of series in A((x)) the series from A((x)) which satisfies the conditions 1)–7) specified below, in which f , g and h denote arbitrary series from A((x)), n and m are arbitrary integers and a, b are arbitrary elements of the ring A. 1) ω(f + g, h) ≡ ω(f , h) + ω(g, h) and ω(f , g + h) ≡ ω(f , g) + ω(f , h). 2) The lowest degree of the series ω(f , g) is not less than the sum of lowest degrees of the series f and g; in addition, the lowest degree of the series ω(x, f ) always exceeds exactly by one the lowest degree of the series f and the lowest degree of the series ω(x −1 , f ) is less exactly by one. 3) ω(1, f ) ≡ f . 4) ω(af , gxn ) ≡ aω(f , g)xn . 5) ω(x n , g) ≡ ω(x, ω(xn−1 , g)) for n > 0 and ω(xn , g) ≡ ω(x −1 , ω(xn+1 , g)) for n < 0. 6) ω(x, ω(x −1 , a)) ≡ a. 7) ω(x −1 , a) = φ−1 (a)x−1 + Δ(a). We denote by φ an extension of the mapping φ to an endomorphism of A+ ((x)) which exists by Proposition 8.9. Similarly, we denote by φ−1 and Δ the same extensions φ and

70 | 9 Laurent rings: examples, relation Δ and we denote by γ(⋅) the endomorphism −Δ(φ−1 (⋅)), then, for the function ω, we have the condition relation +∞

ω(x, a) ≡ ∑ φ(γ i (a))xi+1 , i=0

where the symbol ∑ denotes the above-defined “generalized infinite sum” in the ring A((x)). ◁ In the proof, f and g denote arbitrary series from A((x)), n and m denote two arbitrary integers, and a, b denote two arbitrary elements of the ring A. The “generalized infinite sum” of elements φ(γ i (a))xi+1 in the condition is well defined, since its lowest degree is equal to i + 1. We assume that such a function ω exists, then we prove, by induction, that under these conditions for any positive integer n and all series f in A((x)), we have the following relation: n−1

ω(x, f ) = ω(x, γ n (f )xn ) + ∑ φ(γ i (f ))x i+1 . i=0

(∗)

Indeed, for n = 1, it follows from the condition 4) and the relation ω(x−1 , a) = φ−1 (a)x−1 + Δ(a), which holds by the condition 7), that ω(x−1 , ax) = φ−1 (a) + Δ(a)x. Therefore, ω(x, ω(x−1 , ax)) = ω(x, φ−1 (a)) + ω(x, Δ(a)x). By applying the condition 6), we obtain ax = ω(x, φ−1 (a)) + ω(x, Δ(a)x). Then we set b = φ−1 (a) and obtain ω(x, b) = φ(b)x − ω(x, Δ(φ(b))x) = φ(b)x + ω(x, γ(b)x). By Proposition 8.9, we have the relation ω(x, f ) ≡ φ(f )x + ω(x, γ(f )x),

(∗∗)

as required for the proof of the induction base. Now we assume that the relation (∗) holds for some n. By applying (∗∗) to the relation (∗) for n, we obtain n−1

ω(x, f ) = φ(γ n (f )xn )x + ω(x, γ(γ n (f )x n )x) + ∑ φ(γ i (f ))xi+1 . i=0

9 Laurent rings: examples, relation

| 71

Therefore, by considering the property that γ(gxm ) ≡ γ(g)x m , and the condition 4), we obtain n

ω(x, f ) = ω(x, γ n+1 (f )xn+1 ) + ∑ φ(γ i (f ))xi+1 , i=0

which completes the induction step. Therefore, the relation (∗) is true for all positive integers n. Then we find that, for all positive integers n, we have the inclusion n−1

ω(x, f ) − ∑ φ(γ i (f ))x i+1 ∈ Un+1 ; i=0

this means by the definition of the “generalized infinite sum” that +∞

ω(x, f ) = ∑ φ(γ i (f ))x i+1 . i=0

Now we will construct the function ω, gradually expanding the definition area so that each step is the only possible one. For all series g in A((x)) and all elements b of A, we set ω(1, g) ≡ g, ω(x , b) ≡ φ−1 (b)x−1 + Δ(b). −1

We set also (as shown above, in the only possible way) +∞

ω(x, b) ≡ ∑ φ(γ i (b))xi+1 . i=0

With the use of Proposition 8.9, we extend the definition area of the function ω by the second argument to the whole ring A((x)). Therefore, the function ω(f , g) is defined for all series g in A((x)) and for the series f = x−1 , 1, x. Corresponding to the condition 5), we define inductively for n > 1 ω(xn , g) ≡ ω(x, ω(xn−1 , g)). For n < −1, we set ω(xn , g) ≡ ω(x−1 , ω(xn+1 , g)). We define ω(ax n , g) ≡ aω(x n , g). By Lemma 8.7, we can extend the definition area of ω by the first argument to the whole ring A((x)). It is easy to verify that every construction step is uniquely possible and all required properties of the function ω hold under construction, except, maybe, 6). We prove that we have the condition 6). Indeed, ω(x, ω(x−1 , a)) = ω(x, φ−1 (a)x−1 + Δ(a)) = ω(x, φ−1 (a))x−1 + ω(x, Δ(a)) +∞

+∞

i=0

i=0

= ( ∑ φ(γ i (φ−1 (a)))xi ) + ( ∑ φ(γ i (Δ(a)))xi+1 ) = a. ▷

72 | 9 Laurent rings: examples, relation Proposition 9.3. Let A be a ring and R a set with binary operations of addition and multiplication. Then the following conditions are equivalent. 1) R is a Laurent ring with coefficient ring A. 2) There exist an automorphism φ of the ring A, a homomorphism of Abelian groups Δ: A+ → A+ [[x]], and a bijective mapping π from A((x)) onto R such that the addition in R is defined by the relation π(f ) + π(g) = π(f + g) and the multiplication is defined by the relation π(f )π(g) = π(ω(f , g)), where ω is the function constructed by Lemma 9.2 and based on φ and Δ; in addition, φ and Δ satisfy the relation Δ(ab) = ω(Δ(a), b) + φ−1 (a)Δ(b) for any two elements a, b of A. ◁ 1) ⇒ 2). Indeed, we take a bijective mapping π from A((x)) onto R, as in the first of Definition 6.2 of a Laurent ring. Then we will have the condition the required relation π(f ) + π(g) = π(f + g). We denote by β(⋅, ⋅) the function which maps two series from A((x)) to a series in A((x)) according to the rule β(f , g) ≡ π −1 (π(f )π(g)). Let φ be a twisting automorphism for the element π(x) (i. e., an automorphism of the factor rings A = U0 /U1 induced by the automorphism r → π(x)r(π(x))−1 of the ring U0 preserving U1 ). Then φ−1 coincides with the automorphism of the factor ring U0 /U1 induced by the automorphism r → (π(x))−1 rπ(x). Therefore, for every element a of A, the constant term of the element (π(x))−1 π(a)π(x) coincides with φ−1 (a). Therefore, (π(x)) π(a)π(x) − π(φ−1 (a)) ∈ U1 . −1

By considering properties a, b from 6.2, we obtain π(x−1 )π(a) − π(φ−1 (a)x−1 ) ∈ U0 = π(V0 ). Therefore, the image of the mapping Δ(a) ≡ β(x −1 , a) − φ−1 (a)x −1 is contained in V0 . It is obvious that Δ is a homomorphism of additive Abelian groups. It is directly verified that by properties a and b from 6.2, the function β satisfies all conditions Lemma 9.2 for φ and Δ. Therefore, it follows from the uniqueness that the function ω coincides with the function β. It remains to prove the relation Δ(ab) ≡ ω(Δ(a), b) + φ−1 (a)Δ(b). Since the multiplication in the ring R is associative, we have the relation ω(x−1 , ω(ab)) ≡ ω(ω(x−1 , a), b). By the use of properties of the mapping ω and the relation ω(x−1 , a) ≡ φ−1 (a)x−1 + Δ(a),

9.5 Skew Laurent series with skew derivation |

73

we obtain φ−1 (ab)x−1 + Δ(ab) = ω(Δ(a), b) + ω(φ−1 (a)x−1 , b) = ω(Δ(a), b) + φ−1 (a)(Δ(b) + φ−1 (b)x−1 ). Therefore, the required relation holds. 2) ⇒ 1). To apply Lemma 9.1 to the function ω, we have to verify the relation ω(x, 1) = x,

ω(x−1 , 1) = x−1

and ω(x−1 , ab) ≡ ω(ω(x−1 , a), b).

The first two relations directly follow from condition ω(x −1 , a) = φ−1 (a)x−1 + Δ(a). It remains to prove the third relation. Indeed, ω(x−1 , ab) = φ−1 (ab)x −1 + Δ(ab). On the other hand, ω(ω(x−1 , a), b) = ω(φ−1 (a)x −1 + Δ(a), b) = ω(φ−1 (a)x −1 , b) + ω(Δ(a), b) = φ−1 (a)ω(x−1 , b) + ω(Δ(a), b) = φ−1 (ab)x−1 + φ−1 (a)Δ(b) + Δ(a)b. Therefore, by assumption, we obtain ω(x−1 , ab) = ω(ω(x−1 , a), b). By applying Lemma 9.1 to the function ω, we find that the Abelian group A+ ((x)) with multiplication, defined by the function ω, is a ring. Then the ring is the set R itself with operations of addition and multiplication. The conditions of the first of Definition 6.2 of a Laurent ring directly follow from corresponding properties of the function ω. ▷ Remark 9.4. Although Proposition 9.3 provides a formally complete description of Laurent rings, it does not allow them to be constructed in the general case quite effectively, since checking the required relation for φ and Δ is not much simpler than directly checking the axioms of the ring. Nevertheless, if the image of the homomorphism of Abelian groups Δ is contained in A, then we can use this proposition for the construction of some important examples. In this case, after the redesignation δ = Δ, the condition, imposed on φ and δ in Proposition 9.3, is simplified to the relation δ(ab) = δ(a)b + φ−1 (a)δ(b).

9.5 Skew Laurent series with skew derivation Let A be a ring with automorphism φ and let δ be a φ−1 -derivation (i. e., δ is an endomorphism of the Abelian additive group A+ which satisfies the relation δ(ab) = δ(a)b + φ−1 (a)δ(b) for all a, b in A). By Proposition 9.3, a Laurent ring can be constructed with the use of φ and δ; it is called a skew Laurent series ring with skew derivation. The zero endomorphism δ is a

74 | 9 Laurent rings: examples, relation φ−1 -derivation; in this case, we obtain a skew Laurent series ring. In the case φ = 1A , the φ−1 -derivation turns into an ordinary derivation and the skew Laurent series ring with skew derivation is isomorphic to pseudo-differential operator ring in this case. As an example of a non-trivial φ−1 -derivation with non-identity automorphism φ, we can take the function δ(a) = c(a − φ−1 (a)), where c is some fixed central element of the ring A. The obtained example of a skew Laurent series ring with skew derivation shows that the class of Laurent rings properly contains the union of the class of skew Laurent series rings and the class of pseudodifferential operator rings. Proposition 9.6. Let A be a ring with automorphism φ. Then all conditions of the first of Definition 6.2 of a Laurent ring are satisfied by the Laurent series ring R = A((x, φ)) with the natural isomorphism of the Abelian groups A((x, φ)) and A((x)) which takes the formal sum an xn + an+1 x n+1 + ⋅ ⋅ ⋅ in the first ring to the same formal sum in the second ring. In addition, if an , an+1 , . . . is an arbitrary sequence of elements of the ring A, then a “generalized infinite sum” of monomials ai xi by i ≥ n is always defined and coincides with the formal series an xn + an+1 x n+1 + ⋅ ⋅ ⋅ . Proposition 9.6 is directly verified. Proposition 9.7. Let A be a ring with derivation δ. Then there exists a unique, up to isomorphism, ring A((t −1 , δ)) consisting of formal sums m

f = ∑ fi t i , i=−∞

where t is a variable, m is an integer (maybe negative), and the coefficients fi of the series f are elements of the ring A which satisfy the following properties. 1) The ring A((t −1 , δ)) is a Laurent ring provided we take π: A((x)) → A((t −1 , δ)) as the mapping from the first of Definition 6.2 of a Laurent ring; this mapping takes the −m i i formal sum ∑+∞ i=m fi x to the formal sum of ∑i=−∞ f−i t . −1 2) In the ring A((t , δ)), the relation ta = at + δ(a) holds for every element a of the ring A. In addition, if {an , an+1 , . . .} is an arbitrary sequence of elements of the ring A, then a “generalized infinite sum” of monomials ai x i by i ≥ n is always defined and coincides

9.5 Skew Laurent series with skew derivation | 75

with the formal series an t −n + an+1 t −n−1 + ⋅ ⋅ ⋅ . In addition, we have the relation +∞

t −1 a = ∑ (−1)i δi (a)t −i−1 i=0

in the ring A((t −1 , δ)) for every element a from the ring A. ◁ Proposition 9.7 follows from Proposition 9.3 and the coherence of the generalized infinite sums with formal infinite sums in the ring A((x)) proved in 6.7. ▷ The chosen method of constructing pseudo-differential operator rings does not allow us to define the product of two series by an explicit relation for coefficients; it allows us to prove the existence of such a product by an iterative procedure. Nevertheless, we write an explicit formula, although to use it for verification of some properties of the product is difficult. For the multiplication formula, we need an additional designation. Usually, binomial coefficients (nk ) are defined only for non-negative integers n and non-negative integers k ≤ n, with the use of the formula n n(n − 1)(n − 2) . . . (n − k + 1) , ( )≡ k! k where we assume that 0! = 1. We expand the same definition to all integral values n and all non-negative integral values k. In addition, if k > n ≥ 0, then (nk ) = 0. To prove that, for negative n, the number (nk ) is an integer, we remark that −n (−n)(−n − 1) . . . (−n − k + 1) ( )= k! k n+k−1 k (n)(n + 1) . . . (n + k − 1) = (−1) = (−1)k ( ). k! k Lemma 9.8. Let A be a ring with derivation δ and let A((t −1 , δ)) be the pseudodifferential operator ring. Then, for any integer n and any element a of A, the relation +∞ n t n a = ∑ ( )δi (a)t n−i i i=0

holds in the ring A((t −1 , δ)). ◁ In this proof, the ordinary summation symbol ∑ denotes a “generalized infinite sum” from the condition iii) of the definition 6.3 of a Laurent ring. In addition, we use the property that the formal sum in recording of the elements of the pseudodifferential operator ring is a partial case of a “generalized infinite sum”. For a non-negative integer n in the relation +∞ n t n a = ∑ ( )δi (a)t n−i , i i=0

76 | 9 Laurent rings: examples, relation we can assume that the sum is finite, since for all i > n, the binomial coefficient (ni ) is equal to zero. We prove the required relation by induction on non-negative n. For n = 0, the relation is trivial and for n = 1 it coincides with the relation ta = at + δ(a), which is part of the definition of the pseudo-differential operator ring. Now assume that it is proved for some n; then we will prove it for n + 1. Indeed, we have n n n n t n+1 a = t(t n a) = t ∑ ( )δi (a)t n−i = ∑ ( )(δi (a)t + δi+1 (a))t n−i i i i=0 i=0

n+1 n+1 n n n+1 i = at n+1 + ∑ (( ) + ( ))δi (a)t n+1−i = ∑ ( )δ (a)t n+1−i . i i − 1 i i=1 i=0

It remains to prove the required relation for negative n. For n = −1, it coincides with relation +∞

t −1 a = ∑ (−1)i δi (a)t −i−1 , i=0

which holds by Proposition 9.7. Let us assume that it is proved for some −n, where n is a positive integer. By considering 7.2, we obtain +∞

+∞ −n i −n )δ (a)t −n−i ) = ∑ t −1 ( )δi (a)t −n−i i i i=0

t −n−1 a = t −1 ( ∑ ( i=0

+∞ +∞

−n i+j )δ (a)t −j−1 )t −n−i i

= ∑ ( ∑ (−1)j ( i=0

j=0

+∞ +∞

−n = ∑ ∑ (−1)j ( )δi+j (a)t −n−i−j−1 i i=0 j=0 +∞ i

−n i )δ (a)t −n−i−1 . i−j

= ∑ ∑ (−1)j ( i=0 j=0

By considering the relation (−n ) = (−1)k (n+k−1 ) and by making the replacement j = i−j, k k we obtain +∞

i

i=0

j=0

n+j−1 ))δi (a)t −n−i−1 . j

t −n−1 a = ∑ (−1)i ( ∑ (

By induction on i, it is easy to prove the following arithmetical relation: i

n+j−1 n+i )=( ), j i

∑(

j=0

9.9 An explicit formula for multiplication of two series | 77

which implies the relation +∞ n+i i −n − 1 i )δ (a)t −n−i−1 = ∑ ( )δ (a)t −n−i−1 , i i i=0

+∞

t −n−1 a = ∑ (−1)i ( i=0

as required. ▷ Now we can give an explicit formula for the multiplication of two series.

9.9 An explicit formula for multiplication of two series Let A be a ring with derivation δ and let A((t −1 , δ)) be the pseudo-differential operator ring. For any two elements n

m

f = ∑ fi t i ∈ A((t −1 , δ))

and g = ∑ gi t i ∈ A((t −1 , δ)),

i=−∞

i=−∞

we have the relation n+m

n

fg = ∑ ( ∑

m

i )f δi+j−k (gj ))t k . i+j−k i

∑ (

k=−∞ i=k−m j=k−i

◁ In this proof, we will use an ordinary summation sign ∑ to designate the “generalized infinite sum” from the condition iii) of the second of Definition 6.3 of a Laurent ring. In addition, we use the property that the formal sum in the element record of the pseudo-differential operator ring is a partial case of the “generalized infinite sum”. Indeed, we use 7.2 and the relation proved in Lemma 9.8 to obtain n

m

fg = ( ∑ fi t i )( ∑ gi t i ) i=−∞ n m

= ∑

i=−∞

∑ fi t i gj t j

i=−∞ j=−∞ n

= ∑

m

i

i=−∞ j=−∞ n

= ∑

m

∑

k=−∞

i

n

i )f δi−k (gj )t k+j i−k i

∑ (

i=−∞ j=−∞ k=−∞

= ∑

i )δi−k (gj )t k )t j i−k

∑ fi ( ∑ (

m

i+j

∑

∑ (

i=−∞ j=−∞ k=−∞

i )f δi+j−k (gj )t k . i+j−k i

Assuming temporarily that, for negative k, the binomial coefficient (nk ) is equal to zero, we can extend the upper limit of summation over k to n+m in the last expression (making it not depending on i and j). Then, according to 7.2, we can change the summation

78 | 9 Laurent rings: examples, relation order. We obtain n+m

fg = ∑

n

∑

m

i )f δi+j−k (gj )t k . i+j−k i

∑ (

k=−∞ i=−∞ j=−∞

i ) is equal to zero. This allows one to change lower For i + j − k < 0, the coefficient (i+j−k limits of summation by i and j. We obtain n+m

fg = ∑

n

∑

m

i )f δi+j−k (gj )t k , i+j−k i

∑ (

k=−∞ i=k−m j=−k−i

as required. ▷

9.10 Rings of n-adic integers and fractional n-adic numbers Let n be an arbitrary integer exceeding one. Similar to the ring of p-adic integers, we consider the ring of n-adic integers, i. e., the ring of all sequences of elements a1 ∈ ℤ/nℤ,

a2 ∈ ℤ/n2 Z,

...

such that ak ≡ ak−1 (mod nk−1 ) with term-wise addition and multiplication. It is directly verified that this definition is correct and that the ring of integers ℤ is naturally contained in the ring ℤn (the integer a can be associated with a sequence of images of the number a in the residue ring ℤ/nk ℤ). The ring of n-adic integers is the projective limit of the rings ℤ/nk ℤ. The ring of fractional n-adic numbers ℚn is the ring of fractions of the ring of n-adic integers with respect to the set {1, n, n2 , n3 , . . .}. The ring of p-adic integers is a partial case of the ring of n-adic integers (for prime integer n = p) and the field of p-adic numbers is a partial case of the ring of fractional n-adic numbers. Now we give an example of a generalized Laurent ring which is not a Laurent ring.

9.11 An example of a generalized Laurent ring which is not a Laurent ring For any integer n exceeding one, the ring of fractional n-adic numbers ℤn satisfies conditions i), ii), iii) of the second of Definition 6.3 of a Laurent ring provided we set Uk = nk ℤn , where ℤn is the ring of n-adic integers embedded in Qn . In addition, the ring U0 /U1 is isomorphic to the residue ring ℤ/nℤ and it is not embedded as a unitary subring in the ring Qn . The ring Qn is a field if and only if n = pm , where p is a prime integer and m is a positive integer; Qn even is not a domain for other integers n.

9.11 An example of a generalized Laurent ring which is not a Laurent ring | 79

◁ We prove that the intersection Uk by all integers k consists only of zero, the remaining part of the condition i) is verified trivially. It is sufficient to remark that, for positive integers k, the set Uk ⊂ U0 = ℤn contains only n-adic integers and coincides with the set of all sequences {ai } which satisfy the definition of a n-adic number and for which the first k terms are equal to zero. It follows that the intersection Uk along positive k consists only of zero. The condition ii) is also satisfied, since it suffices to take n ∈ U1 and n−1 ∈ U−1 as mutually inverse elements. Now let uk ∈ Uk , uk+1 ∈ Uk+1 , . . . be the sequence of elements as in the condition iii) of the definition 6.3. We can delete any finite number of initial members of this sequence, determine a “generalized infinite sum” of the remaining elements, and then add previously deleted members to this sum. Therefore, we can assume that k is a nonnegative integer; by adding a finite number of zero terms to the sum, we can make k equal to zero. We now define a “generalized infinite sum” v as the sequence vi ∈ ℤ/ni ℤ, i where vi is equal to ∑i−1 j=0 uj i and uj i ∈ ℤ/n ℤ is the corresponding term of the sequence defining the element uj of the ring of n-adic numbers. It is directly verified that v is the required sum. It follows from the property that U1 contains exactly those sequences whose first term is equal to zero and it is obvious that the ring U0 /U1 is isomorphic to the residue ring ℤ/nℤ. The additive group of the ring Qn is a torsion-free Abelian group, since it contains the field of rational numbers; consequently, the ring U0 /U1 cannot be embedded in it. Now we will prove that the ring of integral (resp., fractional) nm -adic numbers is isomorphic to the ring of integral (resp., fractional) n-adic numbers for any positive integer m. Then we will prove that, for the prime integer p, the ring of fractional pm -adic numbers is a field (it is well known that the ring of fractional p-adic numbers is a field). Indeed, if the sequence a1 ∈ ℤ/nℤ,

a2 ∈ ℤ/n2 ℤ,

...

determines an element a of the ring ℤn , then we can associate to it a subsequence am ∈ ℤ/nm ℤ,

a2m ∈ ℤ/n2 mℤ,

...

of the above sequence which determines an element of the ring ℤnm . Conversely, each sequence a1 ∈ ℤ/nℤ,

a2 ∈ ℤ/n2 ℤ,

...

can be associated with a sequence am ∈ ℤ/nm ℤ,

a2m ∈ ℤ/n2 mℤ,

...

80 | 9 Laurent rings: examples, relation by determining the missing terms from the condition to any sequence ak ≡ ak−1 (mod nk−1 ). It is directly verified that these correspondences determine an isomorphism of the rings. Now let n be a positive integer which exceeds one and is not a power of prime integer. Then we can represent the integer n in the form n = ab, where a, b are coprime positive integers which are not equal to one. We define an n-adic integer v by a sequence vi such that v1 = a and vi is divided by ai in the ring ℤ/ni ℤ. We can construct such a sequence inductively. Indeed, let vi ≡ ai qi (mod ni ) for some integer qi . Since the integers a and bi are co-prime, a is invertible in the residue ring modulo bi and we can find qi+1 such that aqi+1 ≡ qi (mod bi ). Then ai+1 qi+1 ≡ ai qi (mod ai bi ), as required. Similarly, we can define an n-adic integer w so that w1 = b and wi is divided by bi in the ring ℤ/ni ℤ. The elements w and v are non-zero, since w1 and v1 are non-zero. In addition, wv = 0, since for every positive integer i, the element wi vi is divided by ai bi = ni in the ring ℤ/ni ℤ and, consequently, it is equal to zero. Therefore, the ring Qn is not a domain and, as a corollary, Qn is not a division ring. ▷ Remark 9.12. In papers [1, 2, 19, 20, 31, 35, 36], and [37], the rings of skew inverse Laurent series are studied.

10 Noetherian and Artinian Laurent rings It follows from Propositions 9.6 and 9.7 that all results obtained for Laurent rings are passed to skew Laurent series rings and to pseudo-differential operator rings. In skew Laurent series rings, we assume that Un denotes the set of all series which contain the variable with degree not less than n. In the pseudo-differential operator rings, Un denotes the set of all series which contain the variable with degree not exceeding −n. Some other notation and notions are also passed to the skew Laurent series ring and the pseudo-differential operator ring (for example, the notion of the constant term, the notion of the mapping λ, and the notion of the mapping μ). By considering the condition iv) of the definition 6.3 of a Laurent ring from Proposition 7.7, we can obtain a proposition, partial cases of which for the skew Laurent series ring1 and for pseudo-differential operator rings are well known.

10.1 Theorem on Laurent division rings Let A be a ring. 1. If R is a Laurent ring and A coincides with its coefficient ring, then the ring R is a division ring if and only if the ring A is a division ring. 2. If A is a ring with automorphism φ, then the skew Laurent series ring A((x, φ)) is a division ring if and only if its coefficient ring A is a division ring. 3. If δ is a derivation of the ring A, then the pseudo-differential operator ring A((t −1 , δ)) is a division ring if and only if its coefficient ring A is a division ring. ◁ By considering Propositions 9.6 and 9.7, the assertions 2 and 3 are particular cases of the assertion 1. 1. One way to prove the assertion is in Proposition 7.7. Now let R be a division ring. Then the ring A is a domain, since it is embedded in the division ring R. Let a be an arbitrary non-zero element of the ring A. The element π(a) is invertible in the division ring R, let r be its inverse. By the first of Definition 6.2 of a Laurent ring, the series f from A((x)) corresponds to the element r, since r = π(f ). It easily follows from the first definition of a Laurent ring that π(a)r = π(af ). Let fn x n be the lowest term of the series f . Since A is a domain, the element afn xn is non-zero. Then 1 = π(a)r = π(a)π(f ) = π(af ) = π(afn x n + vn+1 ), 1 10.1(2) was already proved in 1.2(7). https://doi.org/10.1515/9783110702248-010

82 | 10 Noetherian and Artinian Laurent rings where vn+1 is contained in Vn+1 . Therefore, we obtain 1 − π(afn x n ) ∈ Un+1 , which is possible only in the case n = 0 and 1 = π(af0 ), since the element afn x n is non-zero. However, then the element a is right invertible, as required. ▷

10.2 Theorem on Noetherian Laurent rings 1.

Let R be a Laurent ring with coefficient ring A. a. The ring R is right Noetherian if and only if the ring A is right Noetherian. b. The ring R is right Noetherian if and only if all cyclic right R-modules are finitedimensional. c. If R is a right semidistributive semilocal ring, then R is a right Noetherian ring. 2. If A is a ring with automorphism φ, then the skew Laurent series ring A((x, φ)) is right Noetherian if and only if its coefficient ring A is right Noetherian; in addition, the ring A((x, φ)) is right Noetherian if and only if all right cyclic R-modules are finite-dimensional. 3. If δ is a derivation of the ring A, then the pseudo-differential operator ring A((t −1 , δ)) is right Noetherian if and only if its coefficient ring A is right Noetherian; in addition, the ring A((t −1 , δ)) is right Noetherian if and only if all right cyclic R-modules are finite-dimensional. ◁ By considering Propositions 9.6 and 9.7, assertions 3 and 2 are particular cases of assertion 1. 1. a. If the ring R is right Noetherian, then the ring A is right Noetherian, since the lattice of right ideals of the ring A is injectively embedded in the lattice of right ideals of the ring R with the use of the mapping μ. Now let A be a right Noetherian ring. We assume that the ring R is not right Noetherian. Then R contains a strictly ascending infinite chain of right ideals B1 , B2 , B3 , . . . . We consider an ascending chain of right ideals λ(B1 ), λ(B2 ), . . . in the ring A. By assumption, the ring A is right Noetherian. Therefore, there exists a positive integer k such that λ(Bn ) = λ(Bk ) for all n > k. However, the ring A is right Noetherian. Therefore, all right ideals λ(Bn ) are finitely generated. Then the right ideal λ(Bk ) is generated by a finite number of elements of the ring A, which are constant terms of elements {ci } from the set U0 ∩ Bk . For any n > k, we apply Lemma 7.6 to the right ideals Bn , Bk and to the family of elements {ci }. Therefore, all right ideals Bn coincide with each other for n > k; this contradicts the assumption. b. All right cyclic modules over the ring R coincide with all factor modules of the module RR . Therefore, if the ring R is right Noetherian, then all right cyclic R-modules are Noetherian, and, consequently, are finite-dimensional.

10.3 Theorem on Artinian Laurent rings | 83

Let all right cyclic R-modules be finite-dimensional. The idea of the remaining part of the proof is taken from [55]. We assume that the ring A is not right Noetherian. Then there exists a strictly ascending chain of right ideals a1 A ⊂ a1 A + a2 A ⊂ . . . . We construct a system of Laurent series f i ∈ A((x)) such that the right ideals π(f i )R form an infinite nonreductable sum of. By considering 3.6(1), this will contradict the assumption. Let g: N → N be some surjective mapping from the set of positive integers onto the set of positive integers such that g −1 (n) is an infinite set for every positive integer n. (We can set, for example, g(n) = n − [√n]2 + 1, where we denote by [x] the integral part of x.) We define a series f i as follows: f2g(n) = an for every n and all remaining coefn ficients of all series f i are equal to the zero. Now we assume that π(f i )R does not form a nonreductable sum. Then there exists an integer i such that π(f i ) ∈ ∑j=i̸ π(f j )R. Therefore, for some family of elements rj from R we have the relation π(f i ) = ∑j=i̸ π(f j )rj and this sum contains only a finite number of non-zero terms. We denote by ℓ the least of the lowest degrees rj . The choice mapping g of the set g −1 (i) is infinite. Therefore, this set contains a positive integer n such that 2n > −ℓ. Then f2in = f2g(n) = an . It follows from the relation π(f i ) = ∑j=i̸ π(f j )rj that n an ∈

∑

n −ℓ j=i,k≤2 ̸

j

fk A.

j

In addition, the coefficient fk can be non-zero only if k is a power of 2 and j

k ≤ 2n − ℓ < 2n+1 . For k equal to 2n , the coefficient fk (for j ≠ i) is equal to j ∑j=i,k 0. Then it follows from the relation (1 + π(fx−1 ))π(gxm ) = 1 ∈ U0 that m = 1 and the product of constant terms of the elements π(f ) and π(x −1 )π(g)π(x) is equal to 1. However, this is impossible, since, by Proposition 8.5, the constant term of the element π(f ) is contained in the radical J(A) and cannot be invertible. This is a contradiction. ▷ Proposition 13.2. Let A be a ring. 1. If R is a Laurent ring with coefficient ring A and the ring R is semilocal, then the ring A is semilocal. 2. If φ is an automorphism of the ring A and the skew Laurent series ring A((x, φ)) is semilocal, then the ring A is semilocal. 3. If δ is a derivation the ring A, and pseudo-differential operator ring A((t −1 , δ)) is semilocal, then the ring A is semilocal. ◁ 1.

Indeed, we assume that the ring A is not semilocal. Then A contains an infinite sequence of maximal right ideals B1 , B2 , B3 , . . . such that the right ideals B1 , B1 ∩ B2 , B1 ∩ B2 ∩ B3 , . . . form a properly descending sequence. By Lemma 8.4, right ideals μ(Bi ) are maximal right ideals of the ring R; in addition, right ideals μ(B1 ), μ(B1 ) ∩ μ(B2 ), μ(B1 ) ∩ μ(B2 ) ∩ μ(B3 ), . . .

also form a strictly descending sequence; this contradicts the property that the ring R is semilocal. 2, 3. The assertions directly follow from 1 and Propositions 9.6, 9.7. ▷ The following simple assertion directly follows from the property that the mapping μ is an injective embedding from the lattice of right ideals of the ring A in the lattice of right ideals of the ring R. Proposition 13.3. Let A be a ring. 1. If R is a right distributive Laurent ring with coefficient ring A, then the ring A is right distributive. 2. If A is a ring with automorphism φ and the skew Laurent series ring A((x, φ)) is right distributive, then the coefficient ring A is right distributive as well. 3. If A is a ring with derivation δ and the pseudo-differential operator ring A((t −1 , δ)) is right distributive, then the coefficient ring A is right distributive as well.

13 Semilocal Laurent rings | 101

Lemma 13.4. Let A be a right distributive ring. 1. If I, J are two right ideals of the ring A, then there no non-zero module homomorphisms between I/(I ∩ J) and J/(I ∩ J). 2. If A right Noetherian, then A is right invariant. 3. If A is a semisimple ring, then A is a finite direct product of division rings. 4. If A is semilocal, then A does not contain infinite nonreductable sums of right ideals. 5. We assume that A is right invariant, the ideal J(A) is nilpotent and the factor ring A/J(A) is a finite direct product of division rings. Then A is the finite direct product of right uniserial right Artinian rings. ◁ 1.

It is sufficient to prove that, if (X ⊕ Y)A is a distributive module and f ∈ HomA (X, Y), then f = 0 for any x ∈ X. By 3.9(1), for elements x ∈ X and f ∈ Y, there exists an element a ∈ A such that xa, f (1 − a) ∈ xA ∩ fA ⊆ X ∩ Y = 0. Then f = f (a + (1 − a)) = fa = f (xa) = f (0) = 0.

2. Let I be a right ideal of the ring A and y ∈ A an arbitrary element of this ring. Since the ring A is Noetherian, the ascending chain I ⊆ (I + yI) ⊆ (I + yI + y2 I) ⊆ ⋅ ⋅ ⋅ stabilizes. Let J = I + yI + ⋅ ⋅ ⋅ + yn I

and

J + yn+1 I ⊆ J.

Then yJ ⊆ J. Let k be the minimal integer such that J = I + yI + ⋅ ⋅ ⋅ + yk I. We assume that yI ⊈ I. Then k is non-zero. Let K = I + yI + ⋅ ⋅ ⋅ + yk−1 I. Then J ≠ K and J = K + yK. By the relation x → yx, a module homomorphism ψ from J in J is defined; it naturally induces the homomorphism ψ from J/K in J/yK. In addition, J/K = (yK + K)/K ≅ yK/(K ∩ yK) and, similarly, J/yK ≅ K/(K ∩ yK). Therefore, by 1, the homomorphism ψ has to be equal to the zero homomorphism. Then ψ(J) = yJ ⊆ yK = ψ(K). Therefore, J = ψ−1 (ψ(J)) = ψ−1 (ψ(K)) = K + ker ψ.

102 | 13 Semilocal Laurent rings Then we denote by K1 the right ideal ψ−1 (K). It is directly verified that J = K + ker ψ ⊆ K + K1 = K1 + ψ(K1 ). By applying to K1 the argument applied earlier to K, we obtain the relation K1 + ker ψ = J. In addition, ker ψ ⊆ K1 . Therefore, ψ−1 (K) = K1 = J. Then yK ⊆ yJ = ψ(J) ⊆ K. Therefore, J = K + yK = K, which contradicts the choice of K. 3. Since the ring A is semisimple, it is a finite direct product of simple rings Ai . By 2, the ring A is right invariant; consequently, every ring Ai is right invariant as well. Any right invariant simple ring is a division ring. Consequently, A is the finite direct product of division rings. 4. Since the ring A is semilocal, there exist only a finite number of non-isomorphic simple right modules over A. Now we assume that S = ∑ Ii is a countable nonreductable sum of non-zero right ideals of the ring A (without loss of generality, we can assume that {Ii } are principal right ideals). We set Jj = ∑i=j̸ Ii . We note that all submodules Jj are not equal to the module S. Every right module S/Jj is a cyclic non-zero module; therefore, it has a simple factor module. Since j runs over an infinite value set and the number of non-isomorphic simple right modules over A is finite, we have at least two isomorphic modules among above-mentioned simple factor modules. This means that there exist distinct subscripts i, j and proper submodules M, N in the module S such that S/M ≅ S/N and Ji ⊆ M, Jj ⊆ N. In addition, S = Ji + Jj ⊆ M + N. Consequently, S/M = (M + N)/M = N/(M ∩ N) 5.

and

S/N = (M + N)/N = M/(M ∩ N).

Therefore, N/(M ∩ N) ≅ M/(M ∩ N), which contradicts the distributivity of A, by 1. Let the factor ring A/J(A) be the finite direct product of division rings Fi . We denote by fi the identity element of the division ring Fi . The identity element of the factor ring A/J(A) is a sum of pairwise orthogonal idempotents fi . A finite system {fi } of pairwise orthogonal idempotents of the factor ring A/J(A) can be lifted to a system of pairwise orthogonal idempotents ei of the ring A, since J(A) is a nil-ideal. Then A = ⨁ ei A. All idempotents of the right invariant ring A are central in A. In addition, Fi = (ei A + J(A))/J(A) ≅ ei A/(ei A ∩ J(A)) is a division ring. Then ei A is a distributive ring (we denote its Ai ) and the ring Ai /(Ai ∩J(A)) is a division ring and the ideal Ai ∩J(A) is nilpotent. The multiplication on ei is a homomorphism from the ring A in the ring Ai . Therefore, Ai ∩ J(A) = ei J(A) ⊆ J(ei A) = J(Ai ),

13 Semilocal Laurent rings | 103

and ei J(A) = J(Ai ), since Ai /ei J(A) is a division ring. Therefore, Ai is a right distributive, right invariant, local ring with nilpotent Jacobson radical. Then the right distributive module J n (Ai )/J n+1 (Ai ) is a vector space over the division ring Ai /J(Ai ) and is simple by distributivity. Therefore, J(Ai )/J 2 (Ai ) = j(Ai /J(Ai )), where j is an arbitrary element of J(Ai )\J 2 (Ai ) and j is its image in the module J(A)/J 2 (A) under the natural epimorphism. We obtain 2

J(Ai ) = jAi + J 2 (Ai ) = jAi + (jAi + J 2 (Ai )) = jAi + j2 Ai + J 3 (Ai ) = ⋅ ⋅ ⋅ = jAi , since J(Ai ) is nilpotent and Ai is invariant. Then J n (Ai ) is generated, as a right ideal, by any element of jn Ai \jn+1 Ai ; this implies that any right ideal of the ring Ai coincides with a power of J(A). Therefore, for every i, the ring Ai is a right uniserial right Artinian ring and the ring A is isomorphic to the direct product of the Ai . ▷ Theorem 13.5. Let R be a Laurent ring with coefficient ring A. Then the following conditions are equivalent. 1) R is a right distributive semilocal ring. 2) R is the finite direct product of right uniserial rings. 3) R is the finite direct product of right uniserial right Artinian rings. 4) A is the finite direct product of right uniserial right Artinian rings Ai and the right ideal μ(Ai ) is a two-sided for all i. ◁ 1) ⇒ 4). It follows from Propositions 13.3 and 13.2 that A is a right distributive semilocal ring. By Lemma 13.4(4), the ring R does not contain an infinite nonreductable sum of right ideals. In addition, it follows from 3.6(1) and Theorem 10.2(1) that A, R are right Noetherian rings. By Lemma 13.4(2), the rings A and R are right invariant. Let N(A) be the prime radical of the ring A. Since A is a right Noetherian ring, the ideal N(A) is nilpotent. Since the ring A is right invariant, N(A) coincides with the set of all nilpotent elements of the ring A. Then A/N(A) is a reduced ring. Since R is a right invariant ring, the ideal μ(N(A)) is two-sided. By Lemma 8.6, the ring R/μ(N(A)) is a Laurent ring with coefficient ring A/N(A). By Lemma 13.1(4), the Jacobson radical J(R/μ(N(A))) is equal to zero; therefore, the Jacobson radical J(R) of the ring R is equal to μ(N(A)). By assumption, the ring R is semilocal; therefore, the ring R/J(R) = R/μ(N(A)) is an Artinian semisimple ring. By Theorem 11.7, the ring A/N(A) is not Artinian. Then its Jacobson radical is nilpotent; consequently, it is equal to zero, since the factor ring A/N(A) cannot have non-zero nilpotent ideals. Therefore, A/N(A) is a semisimple Artinian ring and J(A) = N(A). By applying Lemma 13.4(3) to the ring A/J(A) and by applying Lemma 13.4(5) to the ring A, we find that A is the finite direct product of right uniserial right Artinian

104 | 13 Semilocal Laurent rings rings Ai . Since the ring R is right invariant, all right ideals μ(Ai ) are two-sided, as required. 4) ⇒ 3). Indeed, let the ring A be the direct sum of finite number of rings Ai and let μ(Ai ) be two-sided ideals of the ring R. Since the mapping B → μ(B) preserves finite sums and intersections of right ideals, the ring R is the direct sum of right ideals μ(Ai ) which also are two-sided ideals by assumption. It remains to prove that each of rings μ(Ai ) is a right uniserial right Artinian ring. For this purpose, we prove that the ring μ(Ai ) is a Laurent ring with coefficient ring Ai . Let π be a mapping from A((x)) in R which satisfies conditions 1)–4) of the first of Definition 6.2 of a Laurent ring. A ring Ai ((x)) is naturally embedded in the ring A((x)); therefore, we can consider the restriction πi of the mapping π to Ai ((x)). It is directly verified that the mapping πi carries out the bijection from the ring Ai ((x)) onto the ring μ(Ai ) and satisfies conditions (1)–(4). Therefore, the ring μ(Ai ) is a Laurent ring with coefficient ring Ai and it is a right uniserial right Artinian ring by Theorem 12.2, as required. 3) ⇒ 2). The implication is obvious. 2) ⇒ 1). The ring R is right distributive, since any direct product of right distributive rings is a right distributive ring (this fact is known, it can be verified with the use of 3.9(1)). A ring R is semilocal, since it is a finite direct product of local rings. ▷

13.6 Remarks a. The set of all rational numbers whose denominators are not divided by 2 or 3, is a right distributive semilocal ring, but is not the direct product of right uniserial rings. b. The set A of all rational numbers with odd denominators is a uniserial ring but is not a serial Artinian ring. In particular, A is not a direct product of uniserial Artinian rings. c. It follows from examples a and b that, for an arbitrary ring, the conditions formulated in items 1, 2 and 3 of Theorem 13.5 are not equivalent (although the third condition implies the second condition and the second implies the first condition). d. We note that distributive Laurent series rings are not necessarily semilocal or right (left) Noetherian. We can verify that the required example is the ring A((x)), where A is the direct product of an infinite number of fields Ai , i = 1, 2, 3 . . . . (Indeed, A[[x]] is a commutative distributive ring, since A[[x]] ≅ ∏+∞ i=1 Ai [[x]] and all Ai [[x]] are uniserial rings. Therefore, the ring A((x)) is isomorphic to the ring of fractions of the distributive ring A[[x]] with respect to multiplicatively closed set {1, x, x2 , . . .}.)

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By using Lemmas 11.1(1) and 11.2(1), we can obtain the following Theorems 13.7 and 13.8 as direct corollaries of Theorem 13.5. Theorem 13.7. If A is a ring with automorphism φ, then the following conditions are equivalent. 1) A((x, φ)) is a right distributive semilocal ring. 2) A((x, φ)) is the finite direct product of right uniserial rings. 3) A((x, φ)) is the finite direct product of right uniserial right Artinian rings. 4) A is the finite direct product of right uniserial right Artinian rings Ai and φ(Ai ) = Ai for all i. Theorem 13.8. If A is a ring with derivation δ, then the following conditions are equivalent. 1) The ring A((t −1 , δ)) is a right distributive semilocal ring. 2) The ring A((t −1 , δ)) is a finite direct product of right uniserial rings. 3) The ring A((t −1 , δ)) is a finite direct product of right uniserial right Artinian rings. 4) The ring A is a finite direct product of right uniserial right Artinian rings Ai and δ(Ai ) ⊆ Ai for all i. In connection to Theorem 13.7, we note that the case of Laurent series rings differs from the case of formal power series rings, since we can to verify that the formal power series ring in one variable is a finite direct product of right uniserial rings if and only if the ring of coefficients is a finite direct product of division rings.

13.9 Open questions Let R be a Laurent ring with coefficient ring A. 1. If the ring R is semilocal, then is it true that the Jacobson radical J(R) is a nil-ideal of the ring R? 2. In terms of the ring A, can one find a criterion of the property that the ring R is semilocal? 3. Is it true that R is a right semidistributive semilocal ring if and only if A is a right Artinian right semidistributive ring?

14 Filtrations and (generalized) Malcev–Neumann rings In this section, Malcev–Neumann rings are defined. The class of all Malcev–Neumann rings strongly contains Malcev–Neumann series rings, skew formal Laurent series rings and formal pseudo-differential operator rings. In the section ring-theoretical properties of Malcev–Neumann rings are studied. Malcev–Neumann series rings, skew formal Laurent series rings and formal pseudo-differential operator rings have similar ring-theoretical properties associated with the existence of filtrations with respect to the lowest degree of the series.

14.1 Filtered rings, ordered groups and filtrations Let G be an arbitrary group. The identity element 1G of the group G is simply denoted by 1. A ring R is called a G-filtered ring if R contains a family of additive subgroups {Ug } (where g is an arbitrary element of the group G) such that, for any g and h in G, we have the inclusion Ug Uh ⊆ Ugh and for the identity element 1R of the ring R, we have the inclusion 1R ∈ U1 . In this case, we will also say that the filtration {Ug } is defined on the ring R. The filtration is said to be exhaustive if the union Ug by all g in G is equal to the ring R. The filtration is said to be separable if the intersection Ug through all g from G consists only of zero. We note that U1 always is a unitary subring of the ring R. A group G is said to be ordered if it satisfies the following conditions. 1) The set G is linearly ordered, i. e., all elements of G are mutually ordered with respect to some reflexive anti-symmetrical transitive relation ≤. 2) For any three elements x, y, z ∈ G, the inequality x ≤ y implies the inequalities xz ≤ yz and zx ≤ zy. If there is a filtration {Ug } by an ordered group G on the ring R and for any two elements g, h of G with g ≤ h, we have the inclusion Uh ⊆ Ug , then we say that an ordered filtration is defined on R. It is obvious that, for any g ≥ 1, the set Ug is a two-sided ideal in the ring U1 . For ordered filtrations, we denote by Ug+ the union of the Uh with all h > g. It is obvious that Ug+ ⊆ Ug . Similarly, we denote by Ug− the intersection all Uh by all h < g. It is obvious that Ug ⊆ Ug− . We say that the ordered filtration is a upper strictly exhaustive if the union Ug \Ug+ by all g in G is equal to R except for the zero element. It is easy to see that any upper strictly exhaustive filtration is an exhaustive separable filtration. In the case that G is a free cyclic group, the converse assertion is also true: any exhaustive separable filtration is upper strictly exhaustive. https://doi.org/10.1515/9783110702248-014

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In the case of upper strictly exhaustive filtrations, for any r in R, we denote by deg r the element g ∈ G such that r is contained in Ug \Ug+ . It follows from the definition of upper strictly exhaustive filtrations that, for any non-zero element r of R, the element deg r of G is uniquely defined. We will assume that deg 0 = +∞. The element deg r ∈ G is also called the lowest degree of the element r. It is easy to see that, for any r1 , r2 in R, we have the condition of the inequality deg r1 r2 ≥ deg r1 deg r2 . It follows from the property that the identity element 1R of the ring R is contained in U1 and the inequality deg 1R ≥ deg 1R deg 1R implies that deg 1R = 1. We say that an invertible element y ∈ R is strongly invertible if the relation deg y−1 = (deg y)−1 holds. It is easy to see that, for any strongly invertible element y ∈ R and any element r ∈ R, the relations deg yr = deg y deg r and deg ry = deg r deg y hold. It is directly verified that all strongly invertible elements of a filtered ring form a group.

14.2 Strongly filtered rings, Malcev–Neumann rings and generalized Malcev–Neumann rings Let G be an ordered group and let R be a ring with filtration {Ug } defined by an ordered group G. A ring R is called a Malcev–Neumann ring if the following conditions hold. i) {Ug } is a upper strictly exhaustive filtration. ii) For any g in G, there exists at least one strongly invertible element y ∈ Ug . iii) For any set {rα |α ∈ Ω} of elements of the ring R, there exists at least one element r ∈ R (the generalized sum of elements rα ) such that, if, for some g ∈ G, all elements {rα } (except for a finite number of r1 , . . . , rn ) are contained in Ug , then the difference (r − ∑ni=1 rn ) (difference between the “complete” and a partial sum) is contained in Ug as well. iv) The canonical ring homomorphism from the ring U1 onto its factor ring A = U1 /U1+ splits, i. e., there exists an embedding π: A ⇒ U1 , such that its composition with the canonical homomorphism U1 onto A is the identity automorphism of the ring A. In addition, for any a from A, the relations π(a)R ∩ U1 = π(a)U1 and Rπ(a) ∩ U1 = U1 π(a) hold. A ring which satisfies the conditions i)–ii) but does not necessarily satisfy the conditions iii)–iv), is called a strongly filtered ring or a ring with strong filtration. A ring, which satisfies the conditions i)–iii) but does not necessarily satisfy condition iv), is called a generalized Malcev–Neumann ring.

14.3 Remark on the condition iii) of an infinite summation In this paper, the condition iii) is the main studied property, which allows one to obtain the most non-trivial results. The presence of an “infinite summation” distin-

108 | 14 Filtrations and (generalized) Malcev–Neumann rings guishes series rings from polynomial rings. In the general case, a “generalized sum” from condition (iii) is not uniquely defined.

14.4 Remark on examples of Malcev–Neumann rings We will show below that the class of Malcev–Neumann rings includes Laurent series rings, skew Laurent series rings (where the filtration group ℤ is defined with respect to the lowest degree of the formal series), pseudo-differential operator rings (where the filtration group ℤ is defined by the leading degree of a formal series) and Malcev– Neumann series rings. The notion of a Malcev–Neumann ring allows one to prove the series of general assertions, which will hold for all listed classes of rings.

14.5 The coefficient ring of the ring with ordered filtration It is directly verified that, for any ring R with ordered filtration, U1 is a unitary subring in R and U1+ is a two-sided ideal in U1 . Therefore, we can consider the factor ring A = U1 /U1+ which will be called the coefficient ring of the ring with ordered filtration. Below, we will prove that skew Laurent series rings, pseudo-differential operator rings, and Malcev–Neumann series rings are Malcev–Neumann rings such that the notion of the coefficient ring in these rings coincides with the ordinary definition.

14.6 Some definitions and notation Let R be a ring with ordered filtration and let {Ug } be a family of its subsets, as in the definition. For every element of U1 , its image under the canonical homomorphism U1 onto A = U1 /U1+ is called the constant term of the element. In the case of the Laurent series ring, this definition coincides with a natural definition of the constant term (if we consider the filtration group ℤ with respect to the lowest degree of the series). The constant term is defined only for elements from U1 . The constant term of an element f is denoted by f ̂. It is easy to see that the constant term of the sum (resp., product) of two elements of U1 is equal to the sum (resp., the product) of their constant terms. In a strongly filtered ring, the elements with lowest degree 1 coincide with the elements of U1 with non-zero constant term. It follows from the definition of a strongly filtered ring that every non-zero element f of the ring R can be represented as a product of the form uy, where u ∈ U1 is the element with non-zero constant term and y is a strongly invertible element. For every subset P of the ring R with ordered filtration, we denote by λ(P) the image of the set U1 ∩ P under the canonical homomorphism U1 onto U1 /U1+ (i. e., λ(P) is the set of all constant terms of all elements of U1 ∩ P). It is directly verified that, if P is a left (resp., right) ideal of the ring R, then λ(P) is a left (resp., right) ideal of the coefficient

14.6 Some definitions and notation

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ring A = U1 /U1+ . Therefore, the mapping λ carries out the mapping from the lattice of left (right) ideals of the ring R in the lattice of left (of right) ideals of the ring A, and this mapping preserves the inclusion relation. In addition, if R is a strongly filtered ring and P is its non-zero right (left) ideal, then λ(P) is a non-zero right (left) ideal of the coefficient ring (indeed, if f ∈ P and g = deg f and y is a strongly invertible element from Ug , then fy−1 ∈ U1 ∩ P or y−1 f ∈ U1 ∩ P; in addition, elements fy−1 and y−1 f have non-zero constant terms). If y is a strongly invertible element of the strongly filtered ring R, then the inner automorphism r ⇒ yry−1 preserves the lowest degree of the elements; in particular, it fixes U1 and U1+ . Therefore, it induces the automorphism φy of the coefficient ring U1 /U1+ . We say that it induces the twisting automorphism φy of the coefficient ring A. We prove some auxiliary assertion on strongly filtered rings. Lemma 14.7. Let R be a strongly filtered ring with coefficient ring A such that A is a domain. Then, for any two non-zero elements r1 and r2 of R, the lowest degree of their product is equal to the product of their lowest degrees. ◁ Indeed, if g is the lowest degree of the element r1 and h is the lowest degree of the element r2 , then there exist two elements s1 , s2 of U1 \U1+ such that r1 = yg s1 and r2 = s2 yh , where yg and yh are strongly invertible elements, as in the definition of a strongly filtered ring. Since the elements s1 , s2 have non-zero constant terms, the constant term of their product s1 s2 is non-zero; consequently, it is contained in U1 \U1+ . The product r1 r2 = yg s1 s2 yh is contained in Ugh . We assume, it is also contained in Ugh+ and the product of s1 s2 = yg−1 r1 r2 yh−1 is contained in U1+ , which is impossible. Therefore, the product r1 r2 = yg s1 s2 yh is contained in Ugh \Ugh+ , as required. ▷ For any strongly filtered ring R with group G, the restriction of the mapping r ⇒ deg r to the set of strongly invertible elements R∗ is a homomorphism from the group R∗ onto the group G, and the image of the homomorphism coincides with the whole group G. Therefore, the group G is isomorphic to the factor group of the group R∗ modulo the subgroup R∗1 of all strongly invertible elements of the ring R with lowest degree 1G . Therefore, the structure of the Malcev–Neumann ring on some ring R can be defined not directly through the group G, but by specifying the group of strongly invertible elements R∗ and the set U1 (indicating in this way the subgroup R∗1 ). In addition, it is sufficient to specify not the whole group of strongly invertible elements R∗ but some its “representative subgroup” R/ . We give an exact definition and prove the corresponding assertion. For any strongly filtered ring R with group G, we call the subset R/ ⊂ R a representative group if R/ is a subgroup of the group of strongly invertible elements of the ring R and for every element g of G, the set R/ contains an element of degree g. Remark 14.8. For the Laurent series ring (with filtration by lowest degrees of series) or for the Laurent polynomial ring in one variable x, a representative group always is the set of monomials which are degrees of the variable x with coefficient equal to 1.

110 | 14 Filtrations and (generalized) Malcev–Neumann rings Lemma 14.9. Let R be a ring and R/ a subgroup of the group U(R) of invertible elements of the ring R. Let U1 be a unitary subring in R such that, for every element r of R/ , we have the inclusion r ∈ U1 or r −1 ∈ U1 ; in addition, rU1 r −1 ⊆ U1 . We assume that, for every non-zero r ∈ R among all sets of the form r 󸀠 U1 (where r 󸀠 is an arbitrary element of R/ ) containing r, there exists the smallest set in the sense of ordering with respect to inclusion. Finally, let U1 R/ = R or R/ U1 = R. Then there exist a group G and a strong filtration {Ug } such that U1G = U1 and the set R/ is a representative subgroup of the ring R. ◁ It immediately follows from the relation rU1 r −1 ⊆ U1 that the relations R/ U1 = R, U1 R/ = R, required in the assumption, are equivalent. Let R/1 be the set of elements r ∈ R/ such that r ∈ U1 and r −1 ∈ U1 . It is easy to see that R/1 is a subgroup in R/ . We prove that it is a normal subgroup. Indeed, let r1 be an arbitrary element of R/1 and r an arbitrary element of R. Then rr1 r −1 ∈ rU1 r −1 ⊆ U1 and rr1−1 r −1 ∈ rU1 r −1 ⊆ U1 , i. e., the element rr1 r −1 and its inverse element are contained in U1 . This means that the element rr1 r −1 is contained in R/1 . Then we can consider the factor group G ≡ R/ /R/1 . For every g = rR/1 in the group G, we set Ug ≡ rU1 . This definition is correct (it does not depend on the choice of a specific element r), since rU1 ⊆ rR/1 U1 ⊆ rU1 U1 = rU1 . Let g = rR/1 , h = sR/1 be two arbitrary elements of the group 󸀠 . Then Ug Uh = rU1 sU1 = rs(s−1 U1 s)U1 ⊆ rsU1 U1 = rsU1 = Ugh ; thus, the sets Ug indeed define a filtration on the ring R. For any two elements r, s from R/ , it follows from the assumption that we have the inclusion s−1 r ∈ U1 or the inclusion r −1 s ∈ U1 . In the first case, we obtain r ∈ sU1 ; in the second case, we have s ∈ rU1 . Therefore, for any g, h in G, we see that either Ug is contained in Uh or Uh is contained in Ug . Therefore, the family of sets Ug is linearly ordered with respect to inclusion. It is directly verified that, if Ug = Uh , then g = h; indeed, if r ∈ sU1 and s ∈ rU1 , then s−1 r ∈ U1 and r −1 s ∈ U1 . Therefore, r −1 s ∈ R/1 . Since the family Ug is linear ordered, this allows one to define the corresponding linear order on group G such that the filtration is ordered. Indeed, it is sufficient to prove that g ≤ h if and only if Uh ⊆ Ug . It follows from the relation R/ U1 = R that the constructed filtration is exhaustive. We prove that the constructed filtration is upper strictly exhaustive. Indeed, let r be an arbitrary non-zero element of the ring R. By the assumption, there exists the smallest set, containing r, among all sets Ug (in the sense of the order with respect to inclusion). It is directly verified that the element g of the group G is the lowest degree of the element r (i. e., r ∈ Ug \Ug+ ). It is directly verified that, for every element r 󸀠 of R/ , its lowest degree is its image under the canonical homomorphism R/ onto G. Therefore, all elements R/ are strongly invertible. Therefore, the filtration constructed is strong as well; i. e., each of the sets

14.6 Some definitions and notation

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Ug = r 󸀠 U1 contain at least one strongly invertible element, namely, r 󸀠 . Finally, U1G = U1 by construction. Since the sets Ug were chosen equal to r 󸀠 U1 , we see that the set of all elements r 󸀠 (i. e., R/ ) will be a representative group. ▷ One more auxiliary assertion on a representative group is directly verified. Lemma 14.10. Let R be a strongly filtered ring with filtration {Ug }, where g are elements of the ordered group G and let R/ be a representative group R. Then the set R/1 consists of all elements R/ with lowest degree 1 and it is a normal subgroup in the group R/ , and the factor group R/ /R/1 is isomorphic to G, where we can take as the isomorphism the mapping, which to the element r 󸀠 R/1 matches the lowest degree deg r 󸀠 of the element r 󸀠 . In addition, for elements of R/ , we have the condition that r1󸀠 is contained in r2󸀠 U1 if and only if for their lowest degrees g1 and g2 , we have the inequality g1 ≥ g2 . Now we prove the assertion which shows that the construction of strongly filtered rings allows a subsequent iteration with preserving their properties; thus, the Laurent polynomial ring over a Laurent polynomial ring also is a Laurent polynomial ring in two variables over the original ring. Proposition 14.11. Let S be a strongly filtered ring with coefficient ring R and with filtration {Ug }, where g are elements of an ordered group G. Let R be a strongly filtered ring with coefficient ring A and with filtration {Vh }, where h are elements of an ordered group H. Let S have a representative group S/ in the sense of the filtration {Ug }. In addition, let the set R/ , the image of S1/ (of the set of all elements of S/ with lowest degree 1G ) under the canonical homomorphism onto the coefficient ring R, is a representative group in the sense of the filtration {Vh }. We also assume that, for every s ∈ S/ , the twisting automorphism φs of the ring R, induced by the element s, preserves the lowest degree of any element of R (in the sense of the filtration {Vh }). Then there exists an ordered group G ∘ H which has a normal subgroup X such that X is isomorphic to H, the factor group (G ∘H)/X is isomorphic to G, and the order relation in G∘H is naturally in agreement with the order in G and H. In addition, S is a strongly filtered ring with filtration {W} of the group G∘H and the coefficient ring is isomorphic to A. In addition, all elements of S/ are strongly invertible in the sense of new filtrations {W}. ◁ We define W1 as the set of all elements from U1 whose constant terms (in the sense of the filtration {Ug }) are contained in V1 . We verify all conditions of Lemma 14.9. First, we verify that S = S/ W1 . Indeed, the relation S1/ W1 + U1+ = U1 follows from the property that R/ , the image of S1/ , is a representative group in the ring R. Since the set U1+ is contained in set W1 , we have the relation S1/ W1 = U1 . However, then S = S/ U1 = S/ S1/ U1 W1 = S/ W1 . We also have s󸀠 W1 s󸀠 ⊆ W1 ; this follows from the property, that, for all elements 󸀠 s of S/ , the twisting automorphism φs󸀠 preserves lowest degrees of all elements R. All −1

112 | 14 Filtrations and (generalized) Malcev–Neumann rings elements s󸀠 of S/ have lowest degrees (in the sense of filtrations {Ug }) which either do

not coincide with 1 (in this case, either s󸀠 or s󸀠 is contained in U1+ ⊆ W1 ) or coincides −1 with 1 (in this case, the lowest term s󸀠 or the lowest term s󸀠 is contained in V1 ; this 󸀠 󸀠 −1 also implies that one of the elements s , s is contained in W1 ). It remains to verify that, for every non-zero element s of the ring S, we have the least (in the sense of order with respect to inclusion) set s󸀠 W1 , where the element s󸀠 is contained in S/ . Indeed, since S/ is a representative group in S, the element s can be represented in the form s󸀠1 u1 , where u1 is contained in U1 and has the non-zero constant term u1̂ . In turn, this constant term has a similar representation in the ring R. Therefore, we obtain s ∈ s󸀠1 (s󸀠2 w1 + U1+ ), where w1 is contained in W1 , and the constant term (in the sense of the ring S) of the element w1 has the non-zero constant term (in the sense of the ring R). Therefore, we obtain the required assertion, if we set s󸀠 = s󸀠1 s󸀠2 . Then we can apply Lemma 14.9 and obtain a strong filtration {Wf } and a group F in the ring S such that the set S/ will be a representative group. It remains to verify the required properties of the group G ∘ H ≡ F. Indeed, by Lemma 14.10, the group F is isomorphic to the factor group of S/ modulo the subgroup S1/ 1 consisting of all elements S/ with lowest degree 1 (in the sense of the new filtration {Wf }). By the same lemma, the group G is isomorphic to the factor group S/ modulo the subgroup S1/ and the group H is isomorphic to the factor group of R/ modulo the subgroup R/1 . In addition, we have a chain of embeddings S1/ 1 ⊆ S1/ ⊆ S/ . It remains note that R/ and R/1 are images S1/ and S1/ 1, respectively under the canonical homomorphism U1 onto R. By considering the property that the pre-image of the identity element under this homomorphism is contained in S1/ 1, we obtain the required assertion that G ≅ S/ /S1/ and H ≅ S/1 /S1/ 1. The required properties of the order, defined on this group, are directly verified with the use of Lemma 14.10. ▷ −1

15 Properties of generalized Malcev–Neumann rings For a generalized Malcev–Neumann ring, the following important assertion holds. Lemma 15.1. Let R be a generalized Malcev–Neumann ring, A its coefficient ring, and P a right ideal of R. We assume that there exist elements f1 , f2 , . . . , fn in P ∩ U1 such that constant terms of all elements of P ∩ U1 are contained in the right ideal generated by constant terms of elements f1 , f2 , . . . , fn ; i. e., for every element g of P ∩ U1 , we have the inclusion ĝ ∈ f1̂ A + f2̂ A + ⋅ ⋅ ⋅ + fn̂ A, where fî is the constant term of the element fi and ĝ is the constant term of the element g. Then the right ideal P is generated by n elements f1 , f2 , . . . , fn . ◁ For every g ∈ G, we choose some strongly invertible element yg in Ug which exists by the condition ii) of the definition from 14.2. We denote by Q the right ideal f1 R + f2 R + ⋅ ⋅ ⋅ + fn R of the ring R. Since all elements fi are contained in the ideal P and Q = f1 R + f2 R + ⋅ ⋅ ⋅ + fn R, we have Q ⊆ P. We assume that the assertion of the lemma is not true. Then there exists an element h ∈ P such that h ∈ ̸ Q. Without loss of generality, we can assume that h ∈ U1 (otherwise, we can multiply h by the strongly invertible element ydeg h−1 ). On the set of all possible families {r1 , r2 , . . . , rn }, we define a relation of a partial order. We say that a family {s1 , s2 , . . . , sn } exceeds a family {r1 , r2 , . . . , rn } if and only if the lowest degree gs of the element h − f1 s1 − f2 s2 − ⋅ ⋅ ⋅ − fn sn exceeds the lowest degree gr of the element h − f1 r1 − f2 r2 − ⋅ ⋅ ⋅ − fn rn ; in addition, for any k, we have the inclusion sk − rk ∈ Ugr . We will use the Zorn lemma. For this purpose, we have to prove that any ascending chain has an upper bound. Indeed, let {r1(i) , r(i)2 , . . . , r(i)n }, 0 < i < +∞, be an ascending chain of such families and h(i) = h − f1 r1(i) − f2 r2(i) − ⋅ ⋅ ⋅ − fn rn(i) . Then, for every positive integer i and every k, 1 ≤ k ≤ n, we have, by assumption, the inclusion rk(i+1) − rk(i) ∈ Udeg h(i) . For every k, there exists a “generalized infinite sum”

of elements rk(i+1) − rk(i) for all 1 ≤ i < +∞. Since a “generalized infinite sum” is not necessarily unique, we choose an arbitrary sum. We add rk(1) to this sum and denote by rk the obtained result. We prove then that a family {r1 , r2 , . . . , rn } and is the required upper bound. Indeed, for any k and each i, the element rk − rk(i) , by the definition of a “generalized infinite sum”, has a lowest degree which is not less than the lowest degree of the element rk(i+1) − rk(i) ; this means that the element rk − rk(i) is contained in Udeg h(i) . It remains to prove that, for every i, the lowest degree gr of the element h − f1 r1 − f2 r2 − ⋅ ⋅ ⋅ − fn rn exceeds the lowest degree gr(i) of the element h − f1 r1(i) − f2 r2(i) − ⋅ ⋅ ⋅ − fn rn(i) . https://doi.org/10.1515/9783110702248-015

114 | 15 Properties of generalized Malcev–Neumann rings It is sufficient to prove the non-strict inequality, since the sequence gr(i) strictly increases, and if the element gr is not less than any its term, then it and strictly exceeds each of them. For this purpose, it is sufficient to prove that the lowest degree of the element s = (h − f1 r1 − f2 r2 − ⋅ ⋅ ⋅ − fn rn ) − (h − f1 r1(i) − f2 r2(i) − ⋅ ⋅ ⋅ − fn rn(i) ) is not less than gr(i) . However, s = f1 (r1(i) − r1 ) + ⋅ ⋅ ⋅ + fn (rn(i) − rn ); therefore, the required assertion follows from inclusions fi ∈ U1 and rk − rk(i) ∈ Ug (i) . r It follows from the Zorn lemma that there exists a maximal family {r1 , r2 , . . . , rn }. Since h is not contained in Q = f1 R+f2 R+⋅ ⋅ ⋅+fn R, the element s = h−f1 r1 −f2 r2 −⋅ ⋅ ⋅−fn rn is non-zero. We will find a strongly invertible element y of Udeg s . Then sy−1 is contained in U1 ⊆ P. Then the constant term s0 element sy−1 is contained in λP; therefore, the element s0 of the coefficient ring can be represented in the form f1̂ a1 + f2̂ a2 + ⋅ ⋅ ⋅ + fn̂ an . Let every element ai ∈ A be the constant term of some element ri󸀠 ∈ U1 . Then the constant term of the element sy−1 − f1 r1󸀠 − ⋅ ⋅ ⋅ − fn rn󸀠 is equal to s0 − s0 = 0. Therefore, the lowest degree of the element s − f1 r1󸀠 y − ⋅ ⋅ ⋅ − fn rn󸀠 y = h − f1 (r1 + r1󸀠 y) − ⋅ ⋅ ⋅ − fn (rn + rn󸀠 y) strictly exceeds the lowest degree of y, which is equal to the lowest degree s. By considering the property that ri󸀠 y is contained in Udeg s for all i, we obtain a contradiction to the property that {r1 , r2 , . . . , rn } is the largest family. ▷ Remark 15.2. For left ideals, the analogue of Lemma 15.1 is also true. As a simple corollary of Lemma 15.1, we obtain the following assertion (its partial cases for skew Laurent series rings and pseudo-differential operator rings are well known). Proposition 15.3. Let R be a generalized Malcev–Neumann ring with coefficient ring A. If the constant term of the element r ∈ U1 is right (resp., left) invertible in the ring A, then the element r is right (resp., left) invertible in the ring R. In addition, if A is a division ring, then R is a division ring. ◁ Indeed, let the constant term of the element r be right invertible. Then we apply Lemma 15.1, where we set n = 1, f1 = r, P = R. We obtain R = P = rR, i. e., the element r is right invertible.

15 Properties of generalized Malcev–Neumann rings | 115

Now let A be a division ring. It follows from the above-proved that every element u pf U1 with non-zero constant term is right (and left) invertible. However, every nonzero element r of the ring R can be represented in the form of product uy, where u ∈ U1 has the non-zero constant term and y is a strongly invertible element. Therefore, r is a product of two invertible elements; consequently, it is invertible. ▷ In connection to Proposition 15.3, we note that a Malcev–Neumann ring is a division ring if and only if its coefficient ring is a division ring (see Proposition 16.2). For generalized Malcev–Neumann rings, this is not true, the corresponding example will be constructed below (the ring of fractional pn -adic numbers Qpn , for n > 1, which is a division ring while its coefficient ring is not a domain). To facilitate the construction of examples of Malcev–Neumann rings and generalized Malcev–Neumann rings, we will show that, under the condition iii) of the definition the Malcev–Neumann ring, it is sufficient to verify that there exist “generalized sums” only for countable sets of elements of the special form. Lemma 15.4. Let R be a strongly filtered ring with filtration {Ug }. We assume that, for any infinite sequence of non-zero elements r1 , r2 , . . . such that lowest degrees deg r1 , deg r2 , . . . form a properly ascending sequence, there exists an element r such that, for any positive integer n, the inequality deg r − ∑ni=1 ≥ deg rn+1 holds. Then the element r is a “generalized sum” of elements r1 , r2 , . . . and a “generalized sum” exists for any set of elements in R, i. e., R is a generalized Malcev–Neumann ring. ◁ It is directly verified that the element r is a “generalized sum” of elements r1 , r2 , . . . . We note some general properties of a “generalized sum”. In any strongly filtered ring, if there exists an element a, “generalized sum” of a family of elements of A, and the element b, which is a “generalized sum” of a family of elements B, then there exists at least one “generalized sum” of the united family A ∪ B, which is equal to a + b. Therefore, if we can partition the set of elements on two parts, for each of them a “generalized sum” exists, then it also exists for the whole set. We note also that a “generalized sum” exists for any finite set of elements (it is equal to their ordinary sum). In addition, a “generalized sum” of any number of zeros is equal to the zero; therefore, it is sufficient to prove that there exists a generalized sum only for sets of non-zero elements. A “generalized sum” also exists for any set of elements such that the set of their lowest degrees has no the least element. For example, a “generalized sum” of these elements is equal to the zero. This follows from the following property of such a set A: if almost all elements of A are contained in some filtration set Ug , then Ug contains all elements of the set A. Now let {rα |α ∈ Ω} be an arbitrary family of elements of the ring R. We prove that for the family there exists a “generalized sum”. Without loss of generality, we can assume that all elements rα are non-zero. If the set {rα } can be partition to two subsets, one of which is finite and the second subset satisfies the condition that the set of lowest

116 | 15 Properties of generalized Malcev–Neumann rings degrees does not have the least element, then it is already proved that a “generalized sum” exists. Therefore, we can assume that the set {rα } satisfies the following property: if any finite number of elements are removed from the set, the element with the smallest lowest degree can be always found. Then we construct a chain of elements: r1 is an element with the least lowest degree from all {rα }, r2 is an element with the least lowest degree from the remaining, and so on. The chain of lowest degrees of these elements is monotonously non-decreasing. If this chain stabilizes at the degree g, then we can take the sum of all those elements whose lowest degrees are less than g as a “generalized sum” of the whole set {rα } (there are only finite number of summands in the above sum). If this chain does not stabilize, then we can unite the elements with equal lowest degrees in this chain (we can summarize them, since there are only finite number of elements of every fixed lowest degree) and obtain a chain with properly ascending sequence of lowest degrees and we can apply the assumption to this chain. The remaining part of the assertion is directly verified. ▷ We prove one more important proposition allowing one to construct additional examples of a generalized Malcev–Neumann ring. Proposition 15.5. Under conditions of Proposition 14.11, if the strongly filtered rings S and R are also generalized Malcev–Neumann rings (with respect to filtrations U and V), then the ring S will be a generalized Malcev–Neumann ring with respect to the new filtration W constructed in the proposition. ◁ We use Lemma 15.4. Let s1 , s2 , . . . be a sequence of non-zero elements of the ring S such that the sequence of their lowest degrees f1 , f2 , . . . (in the sense of filtrations {Wf }) strictly increases. Let g1 , g2 , . . . be this sequence of their lowest degrees in the sense of filtrations {Ug }. This sequence is non-decreasing as well, but it can increase non-strictly. If this sequence does not stabilize, it is easy to see that we can take a “generalized sum” of elements s1 , s2 , . . . in the sense of the Malcev–Neumann ring S with filtration {Ug }, it will also be a “generalized sum” in the sense of the Malcev– Neumann ring S with filtration {Wf }. If the sequence g1 , g2 , . . . is stabilized at the element g, then we can remove from the “generalized sum” a finite number of terms with lowest degree less than g and multiply all elements of the chain on one side by some strongly invertible element with lowest degree g −1 . Therefore, we can assume that all members of the sequence s1 , s2 , . . . have the lower degree 1 (in the sense of the filtration U). After this, it suffices to pass to the quotient ring R, consider the corresponding chain r1 , r2 , . . . (image under canonical homomorphism) and obtain its “generalized sum” in the sense of the ring R. Any pre-image of this “generalized sum” in the ring S will be a “generalized sum” of the original elements s1 , s2 , . . . ; this is checked directly. ▷

15 Properties of generalized Malcev–Neumann rings | 117

We can also construct new generalized Malcev–Neumann rings as factor rings of already constructed rings. Proposition 15.6. Let R be a generalized Malcev–Neumann ring with filtration {Ug } and with coefficient ring A. Let I be a two-sided proper ideal of the ring R such that, for any family of elements {rα ∈ I|α ∈ Ω} of this ideal, there exists at least one of its “generalized sums” contained in the same ideal I. Then the factor ring R/I also is a generalized Malcev–Neumann ring with filtration {Vg }, where every filtration set Vg is the image under the canonical homomorphism of the corresponding set Ug . The coefficient ring will be isomorphic to A/λ(I). ◁ We prove that {Vg } is an upper strictly exhaustive filtration. Let s be an arbitrary non-zero element of the ring R/I. It is sufficient to prove that the set of all elements g of the group G with s ∈ Vg has the largest element. Let s = r + I, where r is the element of the ring R not contained in I. Then it is sufficient to prove that the set of all elements g of the group G with r ∈ Ug + I has the largest element or, which is the same, that the set of elements of the form r +j, where j is contained in the ideal I, contains an element with largest lowest degree. We will use the Zorn lemma. Indeed, let r + j1 , r + j2 , r + j3 , . . . be an infinite sequence of elements of the ring R with strictly increasing lowest degrees g1 , g2 , g3 , . . . , and all elements jn are contained in the ideal I. We note that, for any positive integer n, we have the inclusion jn+1 − jn = (r + jn+1 ) − (r + jn ) ∈ Ugn . By assumption, for elements (j2 − j1 ), (j3 − j2 ), . . . , there exists at least one “generalized sum” j contained in I. It is directly verified that the element r + j1 + j has lowest degree exceeding all gn . Thus, it is the required upper bound of the sequence r + j1 , r + j2 , r + j3 , . . . ; this allows one to apply the Zorn lemma. Therefore, it is proved that {Vg } is a upper strictly exhaustive filtration. The remaining conditions of the definition of a Malcev–Neumann ring are verified trivially. Indeed, if R is a strongly invertible element in the ring R with lowest degree g, then r + I is a strongly invertible element in the ring R/I with the same lowest degree, since inclusions r + I ∈ Vg and r −1 + I ∈ Vg −1 hold. If {sα } is a family of elements of the ring R/I and {rα } is the family of their pre-images in the ring R such that the lowest degree of the element sα in the ring R/I coincides with the lowest degree of the element rα in the ring R for all α, then the image of a “generalized sum” of elements rα in the ring R/I is a “generalized sum” of elements sα ; this is directly verified. ▷ Remark 15.7. For Malcev–Neumann rings, the analogue of this assertion is not true. So, we will give below an example of a Laurent series ring with factor ring which is a generalized Malcev–Neumann ring, but is not a Malcev–Neumann ring.

16 Properties and examples of Malcev–Neumann rings Let R be a Malcev–Neumann ring with coefficient ring A. Let π be an embedding of the ring A in the ring R as in the definition of the Malcev–Neumann ring. For every right ideal B of the ring A, we denote by μ(B) the right ideal π(B)R. It follows from condition iv) of the definition that π(B)R ∩ U1 = π(B)U1 ; therefore, λ(μ(B)) = B. The mapping μ carries out the embedding from the lattice of right ideals of the ring A in the lattice of right ideals of the ring R (this embedding is a homomorphism with respect to the lattice operations of addition and intersection, including infinite sums and intersections). It is easy to see that for any principal ideal aA of the coefficient ring we have the relation μ(aA) = π(a)R. Remark 16.1. Unlike the mapping λ, the mapping μ is not defined symmetrically with respect to the right or left multiplication. One could define it for left coefficients, and then it would embed the lattice of left ideals. In addition, it follows from the existence of such a mapping that the condition iv) of the definition the Malcev–Neumann ring holds and the mapping depends on the choice specific bijective mapping π from A in R. Proposition 16.2. If R is a Malcev–Neumann ring with coefficient ring A, then the ring R is a division ring if and only if the ring A is a division ring. ◁ One way, the assertion is proved in Proposition 15.3. Now let R be a division ring. Then the ring A is a domain, since it is embedded in the division ring R. Let a be an arbitrary non-zero element of the ring A. The element π(a) is invertible in the division ring R; let r be its inverse. By Lemma 14.7, we have the relation deg r = deg π(a) deg r = deg 1R = 1. Therefore, we obtain r ∈ U1 , and then we can consider the constant term r ̂ of the element r. By considering the property that the product of constant terms is equal to the ̂ = 1, as required. ▷ constant term of the product, we obtain ar ̂ = 1. Similarly, we have ra For a Malcev–Neumann ring, we can obtain criteria to be an Artinian or Noetherian ring, which also were familiar earlier for skew Laurent series rings, pseudodifferential operator rings, and Malcev–Neumann series rings. Proposition 16.3. Let R be a Malcev–Neumann ring with coefficient ring A. 1. The ring R is right Noetherian if and only if the ring A is right Noetherian. 2. The ring R is right Artinian if and only if the ring A is right Artinian. ◁ 1.

If the ring R is right Noetherian, then the ring A also is right Noetherian, since the lattice of right ideals of the ring A is injectively embedded in the lattice of right ideals of the ring R with the use of mapping μ.

https://doi.org/10.1515/9783110702248-016

16 Properties and examples of Malcev–Neumann rings | 119

Now let A be a right Noetherian ring. We assume that the ring R is not right Noetherian. Then the ring R contains an infinite strictly ascending chain of right ideals B1 , B2 , B3 , . . . . We consider an ascending chain of right ideals λ(B1 ), λ(B2 ), . . . in the ring A. By assumption, the ring A is right Noetherian. Therefore, there exists a positive integer k such that λ(Bn ) = λ(Bk ) for all n > k. However, the ring A is right Noetherian; therefore, all right ideals λ(Bn ) are finitely generated. Then the ideal λ(Bk ) is generated by a finite number of elements of the ring A, which are constant terms of some elements {ci } of the set U1 ∩ Bk . For any n > k, Lemma 15.1 is applicable to the right ideals Bn , Bk and to the family of elements {ci }. Therefore, all right ideals Bn coincide to each other for n > k, which contradicts the choice of the chain B1 , B2 , B3 , . . . . This is a contradiction. 2. The proof similar to the proof of 1. ▷ Remark 16.4. We can note that the proof of Proposition 16.3 uses the condition iv) of the definition the Malcev–Neumann ring only for the proof one way. Therefore, if the coefficient ring A is right Noetherian (resp., right Artinian), then the ring R is right Noetherian (resp., right Artinian), even if R is a generalized Malcev– Neumann ring. Proposition 16.5. Under the conditions of Proposition 15.5, if the generalized Malcev– Neumann rings S and R are also Malcev–Neumann rings (in the sense of filtrations U and V), then, for the new filtration W constructed in the proposition, the ring S will be a Malcev–Neumann ring as well. ◁ The required embedding π from the ring A in the ring S is constructed with the use of simple composition of the embedding π1 from A in R and the embedding π2 from R in S, which exist by the assumption. We have to prove that, for any element a of A, the relations π(a)S ∩ W1 = π(a)W1 and Sπ(a) ∩ W1 = W1 π(a) hold. It is sufficient to prove the first relation, since the second relation is proved similarly. The inclusion π(a)W1 ⊆ π(a)S ∩ W1 is obvious; we prove the converse inclusion. Let s be an arbitrary element from the set π(a)S ∩ W1 . It is contained also in U1 and its image under the canonical homomorphism U1 onto R is contained in π1 (a)R ∩ V1 . By assumption, this means that this image is contained in π1 (a)V1 , i. e., it is of the form π1 (a)v1 . This means that we have the relation s ∈ π(a)π2 (v1 ) + U1+ . Let g be the element s − π(a)π2 (v1 ) of lowest degree (in the sense of the filtration U) and let sg be an arbitrary strongly invertible element with lowest degree g. We note that g strictly exceeds 1. Then the element s󸀠 ≡ (s − π(a)π2 (v1 ))s−1 g is contained in U1 . On the other hand, the same element is contained in π(a)S. Therefore, by assumption, it is contained in π(a)U1 . However, then the element s󸀠 sg is contained in π(a)U1+ ⊆ π(a)W1 . Therefore, the element s = π(a)π2 (v1 ) + s󸀠 sg also is contained in π(a)W1 , as required. ▷

120 | 16 Properties and examples of Malcev–Neumann rings An ordered set is said to be well-ordered below if any its non-empty subsets has the least element.

16.6 Malcev–Neumann series rings One of the main examples of a Malcev–Neumann ring is a Malcev–Neumann series ring (and also Laurent series rings and skew Laurent series rings which are partial cases of Malcev–Neumann series rings). Let A be a ring, G be an ordered group, and let us have mapping φ: G ⇒ Aut(A). For any g ∈ G, we set φg = φ(g). In addition, let us have some mapping ε: G × G ⇒ U(A) which matches some invertible element of the ring A to every pair of elements of the group G. We denote by A((G, φ, ε)) the set of all formal sums of r = ∑ rg g g∈G

(rg ∈ A)

such that the sets Supp r = {g ∈ G | rg ≠ 0} are well-ordered below. On the set A((G, φ, ε)), we can define the structure of a ring by defining the addition as usual and by defining the multiplication by the multiplication rule for monomials: (ag) × (bh) = (aφg (b)ε(g, h))gh. The multiplication can be extended to infinite formal sums of monomials, since the sum, which determines the coefficient of this g, contains only a finite number of nonzero terms. In the general case, it is necessary to impose some restrictions on the mappings φ and ε to guarantee the associativity of this multiplication and the existence of the identity element in the ring. For any g, h, f in the group G and each a in the ring A, the following relations have to hold: (i)

ε(g, h)ε(gh, f ) = φg (ε(h, f ))ε(g, hf );

(ii) φgh (a) = (ε(g, h)) φg (φh (a))ε(g, h). −1

The second condition can be reformulated as follows: φgh = δ(g, h)φg φh , where δ(g, h) is the inner automorphism of the ring A induced by the element ε(g, h).

16.8 The ring A((x, φ)) is a partial case of the Malcev–Neumann series ring

| 121

Proposition 16.7. The set R = A((G, φ, ε)), where φ and ε are defined as in the definition the Malcev–Neumann series ring, is a ring, and its identity element is the monomial 1R = (ε(1, 1))−1 1G , where 1G is the identity element of the group G. Proposition 16.7 is directly verified.

16.8 The ring A((x, φ)) is a partial case of the Malcev–Neumann series ring Let G be equal to the free cyclic group , e(g, h) ≡ 1 and φxn = φn , where φ is a fixed automorphism of the coefficient ring A. Then the conditions of the definition hold automatically and we obtain the skew Laurent series ring A((x, φ)) which is a particular case of a Malcev–Neumann series ring. The above defined notion of a Malcev–Neumann ring is a further generalization of the construction of Malcev–Neumann series rings and also includes pseudo-differential operator rings whose properties are close to the properties of Malcev–Neumann series rings (in particular, to the properties of skew Laurent series rings).

16.9 Malcev–Neumann series rings are Malcev–Neumann rings Let R = A((G, φ, ε)) be a Malcev–Neumann series ring. Then it is a Malcev–Neumann ring and its coefficient ring (as of the Malcev–Neumann ring) is isomorphic to the ring A. ◁ For every element g ∈ G, we denote by Ug the set of formal sums of r ∈ R such that all elements Supp R are not less than g. The condition i) of the definition of a Malcev–Neumann ring is directly verified. To prove that the condition ii) holds, it is sufficient to remark that, for any g in G, the element 1A g is invertible and its inverse element is equal to (ε(g −1 , g))−1 g −1 . It is easy to see that the lowest degree of the element r always coincides with the least element of the set Supp r ⊂ G. We prove that the condition iii) holds. Let {rα |α ∈ Ω} be an arbitrary set of elements of the ring R. We will construct the formal sum of s which is a generalized sum of elements {rα }. For all g in G, such that only a finite number of elements {rα } has the degree which does not exceed g, we set sg equal to the sum of coefficients {rα } of g (this sum is well defined, since only a finite number of these coefficients are non-zero). For all remaining g in G, we set sg = 0. Then the formal sum of s = ∑ sg g will be the required generalized sum of {rα }. We prove this assertion. First, we prove that the formal sum s = ∑ sg g is contained in the ring R; this means that Supp s is well-ordered below set. Indeed, let us assume the contrary. Then Supp s

122 | 16 Properties and examples of Malcev–Neumann rings contains a non-empty subset without the least element. Consequently, we can construct an infinite properly descending chain of elements g1 > g2 > g3 > ⋅ ⋅ ⋅ contained in Supp s. By construction of s, there exist only a finite number of elements rα with lowest degree not exceeding g1 . Let this will be a family r1 , r2 , . . . , rn . Since the coefficient sgi is non-zero for all i and the coefficient sgi is equal to sum of coefficients of gi for all rj , 1 ≤ j ≤ n, we find that, for every positive integer i, there exists at least one rj , such that the coefficient rj of gi is non-zero. Since i runs over all positive integers and j can have only a finite number of values, one of the values j will occur infinitely many times. Without loss of generality, we can assume that this value is j = 1. Then the sequence g1 , g2 , . . . has a subsequence gi1 , gi2 , . . . , for which the coefficient r1 of gik is non-zero for all k. However, then Supp r1 is not a well-ordered below set; this contradicts the inclusion r ∈ R. It remains to prove that the element s satisfies the definition of a generalized sum. Indeed, let g ∈ G and let all elements {rα } (except for a finite number of r1 , . . . , rn ) be contained in Ug . Then the difference n

t = (s − ∑ rn ) i=1

has to be contained in Ug . Indeed, let us assume the contrary. Then the lowest degree t is less than g. Let the lowest degree t be equal to h. The coefficient s of h by construction is equal to the sum of the same coefficients, ∑ni=1 rn . Therefore, the coefficient t of h is equal to zero; this contradicts the choice of h. This contradiction proves that the condition iii) holds. It remains to prove the condition iv) holds. First, we prove that there exists an injective ring homomorphism π from A in R. We denote ε(1, 1) by e. We set π(a) = ae−1 1G . Under this mapping, the identity element 1A of the ring A pass to the identity element 1R = e−1 1G of the ring R. Obviously, we have π(a + b) = π(a) + π(b); we verify that π(ab) = π(a) × π(b). Indeed, π(a) × π(b) = (ae−1 1G ) × (be−1 1G ) = ae−1 φ1 (be−1 )e1G . On the other hand, be−1 1G = 1R × (be−1 1G ) = e−1 φ1 (be−1 )e1G ; therefore, be−1 = e−1 φ1 (be−1 )e. Therefore, π(a) × π(b) = ae−1 φ1 (be−1 )e1G = abe−1 1G = π(ab), as required. It follows from the construction of π that π(A) ∩ U1+ = 0. We also prove that π(A) + U1+ = U1 . Indeed, every element r of U1 can be represented as the sum a1G + s, where

16.10 Skew Laurent series with skew derivation

| 123

a is an element of the ring A and s is contained in U1+ . However, a1G = π(ae), which proves the required relation π(A)+U1+ = U1 . Then we find that the ring A is isomorphic to U1 /U1+ and the required embedding π exists. It remains to verify that π(a)R ∩ U1 = π(a)U1 and Rπ(a) ∩ U1 = U1 π(a). We prove only the first relation, the second relation is proved similarly. Indeed, let s = π(a)r ∈ π(a)R ∩ U1 . Then, if R is this formal sum of monomials rg g, then s is the formal sum of monomials ae−1 φ1 (rg )ε(1, g)g. It follows from the property that s is contained in U1 that all monomials π(a) × rg g = ae−1 φ1 (rg )ε(1, g)g are equal to zero for g < 1. However, this means that 0 = π(a)(r − r 󸀠 ), where r 󸀠 is the formal sum of the monomials rg g with g ≥ 1. In addition r 󸀠 is contained in U1 ; therefore, s = π(a)r 󸀠 ∈ π(a)U1 , as required. ▷

16.10 Skew Laurent series with skew derivation In addition to Malcev–Neumann series rings, skew Laurent series rings with skew derivation, which include skew Laurent series rings and pseudo-differential operator rings, also satisfy the definition of a Malcev–Neumann ring. The formally strict construction of the skew Laurent series ring with skew differentiation is too long, therefore only the statement without proof will be given here. Let A be a ring with automorphism φ and let δ be a φ−1 -derivation (i. e., δ is an endomorphism of the Abelian additive group A+ which satisfies the condition δ(ab) = δ(a)b + φ−1 (a)δ(b) for all a and b from A). Then there exists a unique ring A((x, φ, δ)) such that its additive Abelian group coincides with the additive Abelian group of the Laurent series ring A((x)) and its multiplication satisfies the relations x−1 a = φ−1 (a)x −1 + δ(a),

(ax n )(1xm ) = ax n+m ,

(ax0 )(bxn ) = (ab)xn ;

in addition, the lowest degree of the product of two series is not less than the sum of their lowest degrees. Here, the lowest degree of the series means the degree of the lowest term of the series. It is directly verified that a skew Laurent series ring with skew derivation is a Malcev–Neumann ring, if we take the free cyclic group as the group G and we take as Uxn the set of all series whose lowest degree is not less that n. In addition its coefficient ring, as a Malcev–Neumann ring, is isomorphic to A.

16.11 Iterated Laurent series rings The construction of a Laurent series ring allows for iteration. We can consider the iterated Laurent series ring in n variables A((x1 ))((x2 )) ⋅ ⋅ ⋅ ((xn )) by inductively defining it as the ring ordinary Laurent series in the variable xn with coefficient ring

124 | 16 Properties and examples of Malcev–Neumann rings A((x1 ))((x2 )) ⋅ ⋅ ⋅ ((xn−1 )). Under such a definition, variables are not equitable and their order is important. The properties of iterated Laurent series rings are similar to the properties of the ordinary Laurent series rings; we can show that the iterated Laurent series ring A((x1 ))((x2 )) ⋅ ⋅ ⋅ ((xn )) can be equipped with the structure of a Malcev– Neumann ring such that its coefficient ring is isomorphic to A. The construction of Malcev–Neumann rings can be iterated (subject to certain conditions), as shown in Proposition 16.5.

16.12 Factorization of (generalized) Malcev–Neumann rings Generalized Malcev–Neumann rings can be also obtained by factorization of other Malcev–Neumann rings (or generalized Malcev–Neumann rings). For example, the ring of p-adic integers can be obtained as a factor ring of the formal power series ring ℤ[[x]] modulo the ideal generated by the series (x − p) and the ring of fractional p-adic numbers can be obtained as the factor ring of the Laurent series ring ℤ((x)) modulo the ideal generated by the series x − p. By Proposition 15.6, the ring of fractional p-adic numbers is a generalized Malcev–Neumann ring with coefficient ring ℤ/pℤ. Similarly, we can consider the ring of fractional n-adic numbers for any integer n > 2, which will be a generalized Malcev–Neumann ring with coefficient ring ℤ/nℤ. It is interesting to note the following fact. Proposition 16.13. Let A be a ring, A((x)) the Laurent series ring over A, a some central non-invertible element of the coefficient ring A which is not a zero-divisor, I a two-sided ideal of the ring A((x)) generated by the series x − a, and n an arbitrary positive integer. Then the factor ring A((x))/I is isomorphic to the factor ring A((y))/J, where J is the two-sided ideal of the Laurent series ring A((y)) generated by the series y − an . ◁ We consider a ring homomorphism φ which maps from the ring A((y)) in the ring A((x)): φ maps y to xn and an infinite formal sum of degrees y to the corresponding infinite formal sums of degrees xn . It is clear that φ is a ring monomorphism. It is also obvious that the ideal J passes in the ideal I under the embedding (since its generating element y − an is mapped to the element x n − an contained in the ideal I). Now we prove that the ideal J coincides with pre-image of the ideal I under the homomorphism φ. For this purpose, it is sufficient to prove that the pre-image φ−1 (I) of the ideal I is generated by the central element y − an . By Lemma 15.1, it is sufficient to prove that the set of all constant terms of all series without negative degrees from the pre-image φ−1 (I) is contained in the ideal an A of the ring A. Under the embedding φ, the constant terms series are preserved (and series without negative degrees pass to series without negative degrees); therefore, it is sufficient to prove that all constant terms of all series without negative degrees from the intersection I ∩ φ(A((y))) are contained in an A.

16.12 Factorization of (generalized) Malcev–Neumann rings | 125

We denote by A((xn )) the image φ(A((y))) (since it coincides with the set of all series in A((x)) such that only coefficients of monomials x nk , k ∈ ℤ, are non-zero). Let f be a series without negative degrees contained in I ∩ A((xn )) and f0 its constant term. Then the series f has the form f0 + fn xn + ⋅ ⋅ ⋅ . We assume that f0 is not contained in an A. Let f0 = ak b, where k is some positive integer and the element b of the ring A is not contained in aA. By the assumption, k < n. Then the series f − b(ak − x k ) also is contained in I and its lowest term is equal to bxk and the lowest coefficient is equal to b. However, since the element a is not a zero-divisor, the lowest coefficient of any series from the ideal I = (x − a)A((x)) is contained in the ideal aA. It follows from this contradiction that the ideal J coincides with the pre-image of the ideal I under the homomorphism φ. Now we prove the relation A((x)) = A((x n )) + I. Indeed, every series f = fm x m + fm+1 xm+1 + ⋅ ⋅ ⋅ in A((x)) can be represented in the form of the finite sum of series f (k) = fm+k xm+k + fn+m+k xn+m+k + f2n+m+k x 2n+m+1 + ⋅ ⋅ ⋅ , where k runs through all values up 0 to n − 1. In addition, the series f (k) has the form xk g (k) , where g (k) is a series from A((x n )). Then the series f can be represented in the form f = f0 + xg (1) + x2 g (2) + ⋅ ⋅ ⋅ + xn−1 g (n−1) ∈ f0 + ag (1) + a2 g (2) + ⋅ ⋅ ⋅ + an−1 g (n−1) + I, as required. Now we construct an isomorphism φ between the rings A((y))/J and A((x))/I. If f is a series in A((y)), then we correspond the element φ(f )+I to the element f +J. It follows from the property that the ring homomorphism φ maps from the ideal J in the ideal I that the correspondence φ is well defined and does not depend on the choice of the specific series f . It follows from the property that the ideal J coincides with pre-image of the ideal I that the correspondence φ is one-to-one. It follows from the relation A((x)) = A((xn )) + I = φ(A((y))) + I that the image of the ring homomorphism φ is the whole ring A((x))/I, as required. ▷ It follows from the proved assertion that, for any positive integer k, the field of fractional p-adic numbers is isomorphic to the ring of fractional pk -adic numbers; in addition, these rings are not isomorphic as Malcev–Neumann rings (for example, this follows from the fact that one of the rings has the field ℤ/pℤ as the coefficient ring and the coefficient ring of the ring ℤ/pk ℤ is not a domain for k ≥ 2).

17 Laurent series in two variables It is natural to try to define Laurent series rings in several variables.

17.1 Laurent series rings in several variables Let A be a ring. A Laurent series ring in n variables is the ring A((x1 , x2 , . . . , xn )) consisting of formal sums of the form +∞

+∞

+∞

i

i

f = ∑ ∑ . . . ∑ fi1 i2 ...in x11 x22 . . . xnin , i1 =m1 i2 =m2

in =mn

where m1 , m2 , . . . , mn are arbitrary (maybe negative) integers and the fi1 i2 ...in are arbitrary elements of the ring A. In the ring A((x1 , x2 , . . . , xn )), addition and multiplication are defined in the usual way, by considering the property that variables commute with each other and with coefficients. We note that the variables x1 , x2 , . . . , xn are equitable and any one of their transpositions correctly defines an automorphism of the ring A((x1 , x2 , . . . , xn )). The studies show that the properties of Laurent series rings in two and more variables are very different from the properties of Laurent series rings in one variable. Let A be a ring and A((x, y)) the Laurent series ring in two variables over A. Then we consider the iterated Laurent series rings A((x))((y)) and A((y))((x)). It is easy to see that these two rings do not coincide as the sets of formal sums of degrees x and y (for example, i −i the element ∑+∞ i=0 x y is contained in A((y))((x)) but is not contained in A((x))((y))). In these rings, variables x and y are not equitable and the transposition variables define an isomorphism from A((x))((y)) onto A((y))((x)) and conversely. In addition, the ring A((x, y)) is a subring of the ring A((x))((y)) and A((x, y)) is a subring of the ring A((y))((x)) (moreover, the ring A((x, y)) coincides with the intersection A((x))((y)) and A((y))((x)) as sets of formal sums).

17.2 A remark on lowest terms of Laurent series in several variables In the study of Laurent series in one variable, the lowest term (i. e. the term of the sum containing the smallest degree of the variable) plays an important role. For a series of several variables, the concept of the lowest term needs to be clarified. Considering the non-zero Laurent series in the two variables x and y, we choose from all members of this series those in which x is included in the least degree, and from them we choose the term that contains the least degree of the variable y and call it the lowest term of the series for the permutation (x, y). By interchanging the variables x and y, we get the definition of the lowest term of this series for a transposition (y, x). In the case of https://doi.org/10.1515/9783110702248-017

17.2 A remark on lowest terms of Laurent series in several variables | 127

n variables, for any given permutation of the variables x1 , x2 , . . . , xn , we can define the lowest term. If we fix one permutation of variables, then, obviously, the product of the lowest terms of two series is either zero or equal to the lowest term of the product of these two series An important characteristic of the Laurent series in two variables is the coincidence or mismatch of its two lowest terms. In 10.3, it is proved that a Laurent series ring in one variable is a right Artinian ring if and only if its coefficient ring is a right Artinian ring. The following propositions show that these assertions are not passed to the Laurent series ring in several variables. Proposition 17.3. If A is an arbitrary ring, then the Laurent series ring in two variables A((x, y)) contains a right non-invertible element x + y which is not a left zero-divisor. In particular, A((x, y)) is not a right Artinian ring. ◁ Indeed, we assume that f is some series in A((x, y)) with (x + y)f = 0. Let fxy be the lowest term (x, y) of the series f . Then the lowest term (x, y) of the series (x + y)f has to be equal to yfxy ; consequently, it is non-zero; this contradicts the relation (x + y)f = 0. Now we note that the element x + y is invertible in the ring A((x))((y)), and its inverse is equal to +∞

(x + y)−1 = ∑ (−1)i x−i−1 yi , i=0

but the element (x + y)−1 is not contained in the ring A((x, y)) embedded in A((x))((y)). The inverse element in the ring A((x))((y)) is unique; therefore, the element x + y is not (left or right) invertible in the ring A((x, y)). ▷ Remark 17.4. Considered in the proof of Proposition 17.3, the element x +y is invertible and in the two rings A((x))((y)) and A((y))((x)), but its inverse elements in these rings differ from each other as formal sums of monomials in x and y. Proposition 17.5. If A is an arbitrary ring, then the Laurent series ring in two variables A((x, y)) is not local. ◁ Indeed, if A((x, y)) is a local ring, then either the element (x −1 +y−1 ) is invertible or the element 1 + (x−1 + y−1 ) is invertible. Let one of them (we denote it by f ) be invertible. We note that lowest terms of the series f are equal to fxy = x −1 and fyx = y−1 , respectively. Then let g be the series which is inverse to f and its lowest terms be equal to gxy and gyx (maybe coincident). Then the lowest terms of the series fg are equal to x −1 gxy and y−1 gyx ; they do not coincide, since the degree of x in the first term is at least 1 less than the degree of x in the second term. However, this contradicts the property that fg = 1. ▷ Theorem 17.6. For an arbitrary ring A, the following conditions are equivalent. 1) A((x, y)) is a semiprimitive domain which is not a division ring. 2) The Laurent series ring in two variables A((x, y)) is a domain. 3) A is a domain.

128 | 17 Laurent series in two variables ◁ The equivalence of conditions 3) and 2) easily follows from the property that the lowest term (x, y) product of two series is equal to the product of their lowest terms (x, y) and from the property that the ring A can be naturally embedded in A((x, y)). The implication 1) ⇒ 2) is trivial. 2) ⇒ 1). Indeed, let f be a non-zero series contained in the Jacobson radical of the ring A((x, y)). Then let fxy = ax l yn and fyx = bxk ym its lowest terms (x, y) and (y, x), respectively (in addition, ℓ ≤ k and m ≤ n). If it is necessary, we can replace f by fx−ℓ y−m and we can assume that ℓ = 0, m = 0, fxy = ayn and fyx = bxk . We assume that fxy = fyx , then n = k = 0. Then we replace f by g = f (x + y) and obtain gxy ≠ gyx . Therefore, we can assume that fxy ≠ fyx and n > 0, k > 0. The series f is contained in the Jacobson radical of the ring A((x, y)); therefore, the series 1 + x−1 y−1 f is invertible, but then the series g = xy + f is also invertible. It is easy to see that the lowest terms of the series g are equal to gxy = fxy = ayn

and gyx = fyx = bxk

and, consequently, do not coincide with each other. Let h be the series which is inverse to the series g. Let hxy and hyx be its lower terms (maybe they coincide). Then the lowest terms of the series gh are equal to hxy gxy and hyx gyx , respectively. It is easy to see that the lowest terms of the series gh do not coincide. However, h is inverse to g; therefore, gh = 1. This is a contradiction. It remains to prove that the ring A((x, y)) is not a division ring; this follows from Proposition 17.3. ▷

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Notation A[x, φ] (left) skew polynomial ring [φ, x]A (right) skew polynomial ring A[[x, φ]] (left) skew power series ring [[φ, x]]A (right) skew power series ring A((t −1 , δ)) pseudo-differential operator ring End M endomorphism ring of the module M J(M) Jacobson radical of the module M Sing AA singular ideal of a ring A Sing M singular submodule of the module M U(X) group of invertible elements of a monoid X λ(P) μ(B)

https://doi.org/10.1515/9783110702248-019

8 8 8 7 47 XIV XIV 91 XIII XIII 3, 58 63

Index φ-reduced ring 40 φ−1 -derivation 123 Abelian ring 31 Bezout module XIII Bezout ring XIII biregular ring 40 canonical coefficients of a Laurent series IX coefficient ring of a Laurent ring 49 coefficient ring of A((x, φ)) IX coefficient ring of the ring with ordered filtration 108 coefficients of elements of Laurent rings 63 completely prime ideal XIII constant term in generalized Laurent rings 108 constant term of a Laurent series IX derivation 47 distributive module 22 distributive ring 23 domain XIII essential submodule XIII exhaustive filtration 106 filtered ring 106 filtration 106 finite-dimensional module XIII fully invariant submodule 31 generalized infinite sum 49, 53, 107 generalized Laurent ring 53 generalized left skew Laurent series ring 9 generalized left skew polynomial Laurent ring 9 generalized Malcev–Neumann ring 107 generalized right skew Laurent series ring 9 generalized right skew polynomial Laurent ring 9 Goldie ring XIII inner automorphism 4 invariant module 31 Jacobson radical of the module XIV

Laurent ring 48, 49 Laurent series ring IX leading coefficient of a series 8 leading degree of a series 8 leading term of a series 8 lowest coefficient of a Laurent series IX lowest degree in generalized Laurent rings 57 lowest degree in Malcev–Neumann rings 107 lowest degree of a Laurent series IX lowest term of a Laurent series IX Malcev–Neumann ring 107 nonreductable sum of submodules 20 nonsingular module XIII normal ring 31 normed ring 60 ordered filtration 106 ordered group 106 power series ring IX prime ideal XIII prime radical XIII prime ring XIII principal right ideal ring XIII pseudo-differential operator ring 47 quasi-invariant module 31 reduced ring XII reductable sum of submodules 20 regular module 39 regular ring XII, 39 Rickartian module XII right singular ideal XIII ring of fractional n-adic numbers 78 ring of n-adic integers 78 semilocal module XIII semilocal ring XIV semiprimary ring XIII semiprime ideal XIII semiprime ring XIII semiprimitive module XIV separable filtration 106

136 | Index

serial module XIII simple ring 31 singular submodule of a module XIII skew Laurent series ring IX skew Laurent series ring with skew derivation 73 skew polynomial ring 8 skew power series ring IX skewing automorphism of a generalized Laurent ring 109

strong filtration 107 strong invertibility in Malcev–Neumann rings 107 strongly filtered ring 107 strongly regular module 39 strongly regular ring 39 uniform module XIII uniserial module XIII upper strictly exhaustive filtration 106