Ideal theoretic methods in commutative algebra : in honor of James A. Huckaba's retirement 9780824705534, 082470553X, 9781138401747

Table of contents :
Content: F-rational rings and the integral closures of ideals II
cancellation modules and related modules
abstract ideal theory from Krull to the present
conditions equivalent to seminormality in certain classes of commutative rings
the zero-divisor graph of a commutative ring, II
some examples of locally divided rings
on the dimension of the Jacquet module of a certain induced representation
m-canonical ideals in integral domains II
the t- and v-spectra of the ring of integer-valued polynomials
weakly factorial rings with zero divisors
equivalence classes of minimal zero-sequences modulo a prime
towards a criterion for isomorphisms of complexes
ideals having a one-dimensional fibre cone
recent progress on going-down II
Kronecker function rings -a general approach
on the complete integral closure of the Rees algebra
a new criterion for embeddability in a zero-dimensional commutative ring
finite conductor properties of R(X) and R
building Noetherian and non-Noetherian integral domains using power series
integrality properties in rings with zero divisors
prime-producing cubic polynomials
stability of ideals and its applications
categorically domains - highlighting the (domain) work of James A. Huckaba.

Citation preview

ideal theoretic methods in commutative algebra

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Zuhair Nashed University of Delaware Newark, Delaware

Earl J. Taft Rutgers University New Brunswick, New Jersey

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K.Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Univers itàt Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

N. Jacobson, Exceptional Lie Algebras L-A Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis /. Satake, Classification Theory of Semi-Simple Algebraic Groups F. Hlrzebruch et al., Differentiate Manifolds and Quadratic Forms /. Chavel, Riemannian Symmetric Spaces of Rank One R. B. Burckel, Characterization of C(X) Among Its Subalgebras B. R. McDonald et al., Ring Theory Y.-T. Siu, Techniques of Extension on Analytic Objects S. R. Caradus et al., Calkin Algebras and Algebras of Operators on Banach Spaces E. O. Roxin et al., Differential Games and Control Theory M. Orzech and C. Small, The Brauer Group of Commutative Rings S. Thornier, Topology and Its Applications J. M. Lopez and K. A. Ross, Sidon Sets W. W. Comfort and S. Negrepontis, Continuous Pseudometrics K. McKennon and J. M. Robertson, Locally Convex Spaces M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups G. B. Seligman, Rational Methods in Lie Algebras D. G. de Figueiredo, Functional Analysis L Cesan et al., Nonlinear Functional Analysis and Differential Equations J. J. Schâffer, Geometry of Spheres in Normed Spaces K. Yano and M. Kon, Anti-Invariant Submanifolds W. V. Vasconcelos, The Rings of Dimension Two R. E. Chandler, Hausdorff Compactifications S. P. Franklin and B. V. S. Thomas, Topology S. K. Jain, Ring Theory R R. McDonald and R. A. Mom's, Ring Theory II R. B. Mura and A. Rhemtulla, Orderable Groups J. R. Graef, Stability of Dynamical Systems H.-C. Wang, Homogeneous Branch Algebras E. O. Roxin et al., Differential Games and Control Theory II R. D. Porter, Introduction to Fibre Bundles M. Altman, Contractors and Contractor Directions Theory and Applications J. S. Golan, Decomposition and Dimension in Module Categories G. Fairweather, Finite Element Galerkin Methods for Differential Equations J. D. Sally, Numbers of Generators of Ideals in Local Rings S. S. Miller, Complex Analysis R. Gordon, Representation Theory of Algebras M. Goto and F. D. Grosshans, Semisimple Lie Algebras A. I. Arruda et al., Mathematical Logic F. Van Oystaeyen, Ring Theory F. Van Oystaeyen and A. Verschoren, Reflectors and Localization M. Satyanarayana, Positively Ordered Semigroups D. L Russell, Mathematics of Finite-Dimensional Control Systems P.-T. Liu andE. Roxin, Differential Games and Control Theory III A. Geramita and J. Seberry, Orthogonal Designs J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics C. Byrnes, Partial Differential Equations and Geometry G. Klambauer, Problems and Propositions in Analysis J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields F. Van Oystaeyen, Ring Theory B. Kadem, Binary Time Series J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems R. L. Sternberg et al., Nonlinear Partial Differential Equations in Engineering and Applied Science B. R. McDonald, Ring Theory and Algebra III J. S. Golan, Structure Sheaves Over a Noncommutative Ring T. V. Narayana et al., Combinatorics, Representation Theory and Statistical Methods in Groups T. A. Burton, Modeling and Differential Equations in Biology K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory

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J. Bañas and K. Goebel, Measures of Noncompactness in Banach Spaces O. A. Nielson, Direct Integral Theory J. E. Smith et al., Ordered Groups J. Cronin, Mathematics of Cell Electrophysiology J.W. Brewer, Power Series Over Commutative Rings P. K Kamthan and M. Gupta, Sequence Spaces and Series T. G. McLaughlin, Regressive Sets and the Theory of Isols T. L Herdman et al., Integral and Functional Differential Equations R. Draper, Commutative Algebra W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras R. L Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems J. Van Geel, Places and Valuations in Noncommutative Ring Theory C. Faith, Injective Modules and Injective Quotient Rings A. Fiacco, Mathematical Programming with Data Perturbations I P. Schultz et al., Algebraic Structures and Applications L Bican et al., Rings, Modules, and Preradicals D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces C.-C. Yang, Factorization Theory of Meromorphic Functions O. Taussky, Ternary Quadratic Forms and Norms S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications K. B. Hannsgen et al., Volterra and Functional Differential Equations N. L. Johnson et al., Finite Geometries G. /. Zapata, Functional Analysis, Holomorphy, and Approximation Theory S. Greco and G. Valla, Commutative Algebra A. V. Fiacco, Mathematical Programming with Data Perturbations II J.-B. Hiriart-Urrutyetal., Optimization A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups M. Harada, Factor Categories with Applications to Direct Decomposition of Modules V. I. Istràtescu, Strict Convexity and Complex Strict Convexity V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations H. L Manocha and J. B. Srivastava, Algebra and Its Applications D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems J. W. Longley, Least Squares Computations Using Orthogonalization Methods L. P. de Alcantara, Mathematical Logic and Formal Systems C. E. Aull, Rings of Continuous Functions R. Chuaqui, Analysis, Geometry, and Probability L. Fuchs and L. Salce, Modules Over Valuation Domains P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics W. B. Powell and C. Tsinakis, Ordered Algebraic Structures G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and Their Applications R.-E. Hoffmann and K H. Hofmann, Continuous Lattices and Their Applications J. H. Lightboume III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential Equations C. A. Baker and L M. Batten, Finite Geometries J. W. Brewer et al., Linear Systems Over Commutative Rings C. McCrory and T. Shifrin, Geometry and Topology D. W. Kueke et al., Mathematical Logic and Theoretical Computer Science B.-L. Lin and S. Simons, Nonlinear and Convex Analysis S. J. Lee, Operator Methods for Optimal Control Problems V. Lakshmikantham, Nonlinear Analysis and Applications S. F. McCormick, Multigrid Methods M. C. Tangora, Computers in Algebra D. V. Chudnovsky and G. V. Chudnovsky, Search Theory D. V. Chudnovsky and R. D. Jenks, Computer Algebra M. C. Tangora, Computers in Geometry and Topology P. Nelson et al., Transport Theory, Invariant Imbedding, and Integral Equations P. Clément et al., Semigroup Theory and Applications J. Vinuesa, Orthogonal Polynomials and Their Applications C. M. Dafermos et al, Differential Equations E. O. Roxin, Modern Optimal Control J. C. Diaz, Mathematics for Large Scale Computing

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P. S. Milojevic Nonlinear Functional Analysis C. Sadosky, Analysis and Partial Differential Equations R. M. Shortt, General Topology and Applications R. Wong, Asymptotic and Computational Analysis D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics W. D. Wallis et al., Combinatorial Designs and Applications S. Elaydi, Differential Equations G. Chen et al., Distributed Parameter Control Systems W. N. Everitt, Inequalities H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differential Equations O. Anno et al., Mathematical Population Dynamics S. Coen, Geometry and Complex Variables J. A. Goldstein et al., Differential Equations with Applications in Biology, Physics, and Engineering S. J. Andima et al., General Topology and Applications P Clément et ai, Semigroup Theory and Evolution Equations K. Jarosz, Function Spaces J. M. Bayod et al., p-adic Functional Analysis G. A. Anastassiou, Approximation Theory R. S. Rees, Graphs, Matrices, and Designs G. Abrams et al., Methods in Module Theory G. L Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing M. C. Joshi and A. V. Balakrishnan, Mathematical Theory of Control G. Komatsu and Y. Sakane, Complex Geometry /. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations T. Mabuchi and S. Mukai, Einstein Metrics and Yang-Mills Connections L Fuchs and R. Gôbel, Abelian Groups A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum G. Dore et al., Differential Equations in Banach Spaces T. West, Continuum Theory and Dynamical Systems K. D. Bierstedtetal., Functional Analysis K. G. Fischer et al., Computational Algebra K. D. Elworthyetal., Differential Equations, Dynamical Systems, and Control Science P.-J. Cahen, et al., Commutative Ring Theory S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions P. Clément and G. Lumer, Evolution Equations, Control Theory, and Biomathematics M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research W. O. Bray et al., Fourier Analysis J. Bergen and S. Montgomery, Advances in Hopf Algebras A. R. Magid, Rings, Extensions, and Cohomology N. H. Pavel, Optimal Control of Differential Equations M. Ikawa, Spectral and Scattering Theory X. Liu and D. Siegel, Comparison Methods and Stability Theory J.-P. Zolésio, Boundary Control and Variation M. Kriieketai, Finite Element Methods G. Da Prato and L. Tubaro, Control of Partial Differential Equations E. Ballico, Projective Geometry with Applications M. Costabelet ai, Boundary Value Problems and Integral Equations in Nonsmooth Domains G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations S. Huggett, Twistor Theory H. Cook et ai, Continua D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings K. Jarosz, Function Spaces V. Ancona et ai, Complex Analysis and Geometry E. Casas, Control of Partial Differential Equations and Applications N. Kalton et ai, Interaction Between Functional Analysis, Harmonic Analysis, and Probability Z. Deng et ai, Differential Equations and Control Theory P. Marcellini et ai Partial Differential Equations and Applications A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type M. Maruyama, Moduli of Vector Bundles A. Ursini and P. Aglianò, Logic and Algebra X. H. Cao et ai, Rings, Groups, and Algebras D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models

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J. E. Andersen et al., Geometry and Physics P.-J. Cahen et al., Commutative Ring Theory J. A. Goldstein et al., Stochastic Processes and Functional Analysis A. Sorbi, Complexity, Logic, and Recursion Theory G. Da Prato and J.-P. Zolésio, Partial Differential Equation Methods in Control and Shape Analysis D. D. Anderson, Factorization in Integral Domains N. L Johnson, Mostly Finite Geometries D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm-Liouville Problems W. H. Schikhofet ai, p-adic Functional Analysis S. Sertoz, Algebraic Geometry G. Caristi and E. Mitidieri, Reaction Diffusion Systems A. V. Fiacco, Mathematical Programming with Data Perturbations M. Kfizek et al., Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Estimates S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups V. Drensky et al., Methods in Ring Theory W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions P. E. Newstead, Algebraic Geometry D. Dikranjan and L Salce, Abelian Groups, Module Theory, and Topology Z. Chen et al., Advances in Computational Mathematics X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs C. Y. Yildirim and S. A. Stepanov, Number Theory and Its Applications D. E. Dobbs et al., Advances in Commutative Ring Theory F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry J. Kakol et al., p-adic Functional Analysis M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups F Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of Algebras R. Costa et al., Nonassociative Algebra and Its Applications T.-X. He, Wavelet Analysis and Multiresolution Methods H. Hudzik and L Skrzypczak, Function Spaces: The Fifth Conference J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis G. Lumerand L Weis, Evolution Equations and Their Applications in Physical and Life Sciences J. Cagnol et al., Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G Chen et al., Control of Nonlinear Distributed Parameter Systems F. Ali Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra Á. Granja et al., Ring Theory and Algebraic Geometry

Additional Volumes in Preparation

ideal theoretic methods in commutative algebra in h o n o r o f J a m e s A . H u c k a b a ' s r e t i r e m e n t

e d i t e d by Daniel D . A n d e r s o n

University of Iowa Iowa City, Iowa

Ira J . Papick

University of Missouri Columbia, Missouri

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 First issued in hardback 2019 © 2001 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN-13: 978-0-8247-0553-4 (pbk) ISBN-13: 978-1-138-40174-7 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com!) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Preface

The conference Ideal Theoretic Methods in Commutative Algebra was convened to honor Professor James A. Huckaba's retirement from the Mathematics Department at the University of Missouri-Columbia. This Proceedings consists of research papers presented at that conference, as well as other contributed manuscripts. The combined effort reflects the substantial work of 38 authors and sharply illustrates the diversity of this field of study. Jim received his Ph.D. in mathematics from the University of Iowa in 1967. In that same year he came to the University of Missouri, and from that point on, he was consistently one of the Department's most valued members. During his 32-year academic career, he published numerous research papers in highly regarded mathematics journals, and authored and co-authored two research-level books in commutative ring theory (Commutative Rings with Zero Divisors, Marcel Dekker, 1977; Prüfer Domains, Marcel Dekker, 1997). Jim's enthusiasm and success in research was clearly reflected in his distinguished classroom teaching. His abilities extended past the classroom to seminars and colloquia, where he inspired both students and faculty. In particular, he successfully directed six Ph.D. students in commutative algebra, namely: 1. James Hays, Reduction of Ideals in Commutative Rings, 1971; Paul A. Froeschl III, Chained Rings and Maximal Conductor Rings, 1974; George W. Hinkle, Generalized Kronecker Function Rings and the Ring R(X), 1975; James M. Keller, Topics in the Theory of Graded Rings, 1978; Thomas G. Lucas, The Annihilator Conditions Property (A) and (AC), 1983; Albert Dixon, A Polynomial Ring Localization, 1987. Jim Huckaba is a rare individual of intelligence and compassion, and his professional contributions and accomplishments signify a life of outstanding academic achievement. The enormous success of this conference and the eminent quality of these Proceedings is directly due to the exhilarating talks and superb contributed papers of our colleagues. We extend our sincere appreciation to all participants. We are also thankful to the University of Missouri Mathematics Department and its Chair, Elias Saab, for supporting, both financially and philosophically, the conference and the retirement banquet. Further recognition is extended to Daniel Lieman for his notable contributions to the organization of the conference and banquet. His efficient style, coupled with his careful attention to details, was greatly appreciated. We are especially grateful to Amy Jo Wright, Wendy Hodge, and Brenda Frazier (the exceptional secretarial staff of UMC Mathematics Department) for their professional assistance in the preparation and handling of the conference and banquet. It was a wonderful joy to work with such dedicated and diligent individuals. Additionally, we are deeply indebted to Brenda Frazier, an extremely talented technical typist, computer stylist, and photographer, for iii

iv

Preface

her masterful presentation of conference and banquet materials, as well as her expert and efficient editing assistance on this volume. Two more acknowledgements are appropriate. The inspired poetry and melodic song of Ann Marie McGarry-Papick brought the conference banquet festivities to a new level of artistic appreciation. Finally, and most importantly, I would like to thank Bev Huckaba for helping us all express our respect and affection for Jim Huckaba. Daniel D. Anderson Ira J. Papick

Contents Preface Contributors

Hi vii

1.

F-Rational Rings and the Integral Closures of Ideals II Ian M. Aberbach and Craig Huneke

2.

Cancellation Modules and Related Modules D. D. Anderson

13

3.

Abstract Ideal Theory from Krull to the Present D. D. Anderson and E. W. Johnson

27

4.

Conditions Equivalent to Seminormality in Certain Classes of Commutative Rings David F. Anderson andAyman Badawi

5.

The Zero-Divisor Graph of a Commutative Ring, II David F. Anderson, Andrea Frazier, Aaron Lauve and Philip S. Livingston

61

6.

Some Examples of Locally Divided Rings Ayman Badawi and David E. Dobbs

73

7.

On the Dimension of the Jacquet Module of a Certain Induced Representation William Banks, Daniel Bump, and Daniel Lieman

8.

m-Canonical Ideals in Integral Domains II Valentina Barucci, Evan Houston, Thomas G. Lucas, and Ira J. Papick

9.

The t- and v-Spectra of the Ring of Integer-Valued Polynomials Over a Valuation Domain Paul-Jean Cahen, Evan Houston, and Francesca Tartarone

10.

Weakly Factorial Rings with Zero Divisors Gyu Whan Chang

119

11.

Equivalence Classes of Minimal Zero-Sequences Modulo a Prime Scott T. Chapman, Michael Freeze, and William W. Smith

133

12.

Towards a Criterion for Isomorphisms of Complexes S. Dale Cutkosky and Hema Srinivasan

147

13.

Ideals Having a One-Dimensional Fiber Cone Marco D'Anna, Anna Guerrieri and William Heinzer

155

14.

Recent Progress on Going-Down II David E. Dobbs

171

v

1

49

85 89

109

vi

Contents

15.

Kronecker Function Rings: A General Approach Marco Fontana and K. Alan Loper

189

16.

On the Complete Integral Closure of the Rees Algebra Stefania Gabelli and Anna Guerrien

207

17.

A New Criterion for Embeddability in a Zero-Dimensional Commutative Ring Robert Gilmer

18.

Finite Conductor Properties of R(X) and R Sarah Glaz

19.

Building Noetherian and Non-Noetherian Integral Domains Using Power Series William Heinzer, Christel Rotthaus, and Sylvia Wiegand

20.

Integrality Properties in Rings with Zero Divisors Thomas G. Lucas

265

21.

Prime-Producing Cubic Polynomials Joe L. Mott and Kermit Rose

281

22.

Stability of Ideals and Its Applications Bruce Olberding

319

23.

Categorically Domains: Highlighting the (Domain) Work of James A. Huckaba Ira J. Papick

Index

223 231

251

343 357

Contributors Ian Aberbach, Department of Mathematics, University of Missouri, Columbia, MO 65211: [email protected] Daniel D. Anderson, Department of Mathematics, University of Iowa, Iowa City, IA 52242: dan-anderson @uiowa.edu David F. Anderson, Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300: [email protected] Ayman Badawi, Department of Mathematics and Computer Science, Birzeit University, P.O. Box 14, Birzeit WestBank, Palestine, via Israel: [email protected] William Banks, Department of Mathematics, University of Missouri, Columbia, MO 6521 l:[email protected] Valentina Barucci, Dipartimento di Matemática, Universita di Roma "La Sapienza", 00185 Roma, Italy: [email protected] Daniel Bump, Department of Mathematics, Stanford University, Palo Alto, CA 94305: [email protected] Paul-Jean Cahen, Faculte des Sciences de Saint Jerome, 13397 Marseille cedex 20 France: [email protected] Gyu Whan Chang, Department of Mathematics, Pohang University of Science and Technology, Pohang, 790-784, Korea: [email protected] Scott Chapman, Department of Mathematics, Trinity University, San Antonio, TX 78212-7200: [email protected] S. Dale Cutkosky, Department of Mathematics, University of Missouri, Columbia, MO 65211: dale ©cutkosky.math.missouri.edu Marco D'Anna, Universita di Catania-Dipartimento di Matemática, Viale Andrea Doria, 6-95125 Catania, Italy: [email protected] David E. Dobbs, Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300: [email protected] Marco Fontana, Dipartimento di Matemática, Universita degli Studi Roma Tre, Largo San Leonardo Murialdo, 1, 00146 Roma, Italy: [email protected] Andrea Frazier, Mathematics Department, University of Iowa, Iowa City, IA 52242: [email protected] vii

viii

Contributors

Michael Freeze, Department of Mathematics and Statistics, University of North Carolina at Wilmington, Wilmington, NC 28403-2870:[email protected] Stefania Gabelli, Dipartimento di Matemática, Universita degli Studi Roma Tre, Largo San Leonardo Murialdo, 1, 00146 Roma, Italy: [email protected] Robert Gilmer, Department of Mathematics, Florida State University, Tallahassee, FL 32306-3027: [email protected] Sarah Glaz, Department of Mathematics, University of Connecticut, Storrs, CT 06269: [email protected] Anna Guerrieri, Universita di L'Aquila-Dipartimento di Matemática, Via Vetoio, Loc. Coppito - 67100 L'Aquila, Italy: [email protected] William Heinzer, Department of Mathematics, Purdue University, West Lafayette, IN 47907: [email protected] Evan Houston, Mathematics Department, University of North Carolina-Charlotte, Charlotte,NC 28223: [email protected] Craig Huneke, Department of Mathematics, University of Kansas, Lawrence, KS 66045: [email protected] E.W. Johnson, Department of Mathematics, The University of Iowa, Iowa City, IA 52242: [email protected] Aaron Lauve, Mathematics Department, Rutgers University, New Brunswick, New Jersey 08903: [email protected] Daniel Lieman, Department of Mathematics, University of Georgia, Athens, GA 30602: [email protected], and Department of Mathematics, University of Missouri, Columbia, MO 65211 : [email protected] Philip S. Livingston, Social Work Office of Research and Public Service, The University of Tennessee, Knoxville, TN 37996: plivingl @utk.edu K. Alan Loper, Department of Mathematics, Ohio State University-Newark, Newark, OH 43055: [email protected] Thomas G. Lucas, Department of Mathematics, University of North Carolina-Charlotte, Charlotte, NC 28223-0001: tglucas @ email .unce .edu Joe Mott, Department of Mathematics, Florida State University, Tallahassee, FL 323063027: [email protected]

Contributors

ix

Bruce Olberding, Department of Mathematics, Northeast Louisiana University of Louisiana at Monroe, Monroe, LA 71209: [email protected] Ira J. Papick, Department of Mathematics, University of Missouri, Columbia, MO 65211 : [email protected] Kermit Rose, Computer Center, Florida State University, Tallahassee, Florida (retired): [email protected]. Christel Rotthaus, Michigan State University, East Lansing, MI 488241027 :rotthaus@math. msu.edu William W. Smith, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250: [email protected] Hema Srinivasan, Department of Mathematics, University of Missouri, Columbia, MO 65211: [email protected] Francesca Tartarone, Universita di Trieste, Piazzale Europa 1, Trieste, Italy: [email protected] Sylvia Wiegand, Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0323: [email protected]

James A. Huckaba

F-RATIONAL RINGS AND THE INTEGRAL CLOSURES OF IDEALS II IAN M. ABERBACH*, Department of Mathematics, University of Missouri, Columbia, MO 65211, [email protected] CRAIG HUNEKE*, Department of Mathematics, University of Kansas, Lawrence, KS 66045, [email protected]

Dedicated with fondness to our colleague Jim Huckaba upon the occasion of his retirement.

1

INTRODUCTION

Let R be a commutative Noetherian ring and / an ideal in R. There are many natural and important ways in which the study of the powers of I arises, e.g., from the study of Hilbert functions or the blowing up of / . In studying the powers of i" one is soon led to consider ideals related to J, but simpler, which still reflect asymptotically the behavior of In as n —• oo. Northcott and Rees introduced the ideal of a reduction of an ideal in [22]. If J Ç / , then J is a reduction of I if there exists r such that / r + 1 = J F', and J is a minimal reduction if J is minimal with respect to inclusion among reductions. When (R, m) is local and |i2/m| = oo then every minimal reduction has the same number of generators. This number, £(I), is the analytic spread of J and lies in between ht(J) and dimi?, so is not too big. Moreover, J and / have the same integral closure, where the integral closure is I = {x e R | xn -f r\xn~l + • • • + r n = 0, with r¿ e ft}. It is natural to consider the relationship between the integral closures of powers of / and (powers of) a reduction J. One such relationship has come to be known as the "Briançon-Skoda theorem" [6]. When (R, m) is regular, / has analytic spread £, and J is a reduction of i* then for all w > 0, ft+w € Jw+1 ([20]). The present authors have written a series of papers refining this theorem [2]-[4], [5], part of a renewed interest in furthering the results. Also see [15, 16, 18, 19]. We give a more extensive history of theorems of Briançon-Skoda type in §2 below. Many of the results in [2]-[4] are proved using tight closure [9], which we now describe. *Both authors were partially supported by the NSF. The first author was partially supported by the MU Research Board

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Aberbach and Huneke

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Let fíbea Noetherian ring of positive prime characteristic p > 0. We use q for a varying power of p. Let fl° be the complement of the minimal primes of R and /to! = (iq :ie I). The tight closure of I is I* = {x e R : for some c 6 R°. cxq G J' 9 ' for g > 0}. If I* — I then / is said to be tightly closed. A ring in which every ideal is tightly closed is called weakly F-regular. An ideal I = (xi,... ,xn) is called a parameter ideal if ht (J) > n. R is called F-rational if every parameter ideal of R is tightly closed. K.E. Smith has shown that an F-rational ring has rational singularities [24]. Conversely, Hara [8], and independently Mehta and Srinivas [23] that if X is a scheme of finite type over a field of characteristic 0, then if X has rational singularities it has F-rational type. 2

T H E O R E M S OF B R I A N Ç O N - S K O D A T Y P E

The original theorem of Briançon and Skoda stems from the observation than in On = C { ¿ i , . . . , zn}, the ring of convergent power series, if / G On is a non-unit then / is integral over / = (•${-, • • , 37-)• I n particular, there exists t such that fl e I. Mather asked if there was a uniform value of t sufficing for all non-units / . This question was answered affirmatively by Briançon and Skoda using a deep transcendental result of Skoda's: THEOREM 2.1 (Briançon-Skoda, 1974). Let I C On be an ideal which can be generated by i elements. Then for all w > 0, je+w c

iw+l.

In particular, an answer to the above question is that t — n works for On. The statement of this theorem is completely algebraic, and M. Hochster suggested that the commutative algebra community should provide an algebraic proof. This proof was provided by Lipman and Sathaye for regular rings, and a partial answer was given by Lipman and Teissier for pseudo-rational rings. THEOREM 2.2 (LS). Let (fl, m) be a regular local ring and suppose that I is generated by I elements. Then for all w > 0, Jt+W Q JW + 1,

THEOREM 2.3 (LT). Let (fl, m) be a ring which has only pseudo-rational singularities when localized at any prime ideal of fl. Fix n. Say J Ç I is a reduction such that aim RP < S for all P associated to Jn. Then Jn+*-i Ç Jn. In particular, if J is generated by a regular sequence this holds for all n with ¿ = ht(J). The proofs of Theorems 2.2 and 2.3 are quite involved. Hochster and Huneke gave a generalization for rings containing a field, which says that for an ideal / with an ¿-generated reduction J, I£+w Ç (J™ +1 )*. The proof in characteristic p is elementary. As regular rings are weakly F-regular, this gives the Briançon-Skoda theorem for regular rings containing a field. This approach has been generalized by I. Swanson to joint reductions in [25, 26]. If one examines the proof that Ie+W Ç (J™*1)* (taking w = 0 for simplicity), given z e Ie, one constructs an element c G R° such that czq G (Je)q- As J is generated by t elements, this implies that czq € J[