Rings, Modules, Algebras, and Abelian Groups

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Rings, Modules, Algebras, and Abelian Groups Proceedings of the Algebra Conference—Venezia

edited by

Alberto Facchini University of Padua Padua, Italy

Evan Houston University of North Carolina at Charlotte Charlotte, North Carolina, U.S.A.

Luigi Salce University of Padua Padua, Italy

MARCEL DEKKER, INC.

NEW YORK • BASEL

Although great care has been taken to provide accurate and current information, neither the author(s) nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage, or liability directly or indirectly caused or alleged to be caused by this book. The material contained herein is not intended to provide specific advice or recommendations for any specific situation. Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-4807-7 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016, U.S.A. tel: 212-696-9000; fax: 212-685-4540 Distribution and Customer Service Marcel Dekker, Inc. Cimarron Road, Monticello, New York 12701, U.S.A. tel: 800-228-1160; fax: 845-796-1772 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-260-6300; fax: 41-61-260-6333 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright © 2004 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

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

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

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 Universitat 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. Hirzebruch et a/., Differentiable Manifolds and Quadratic Forms I. Chavel, Riemannian Symmetric Spaces of Rank One R. B. Burckel, Characterization of C(X) Among Its Subalgebras B. R. McDonald et a/., Ring Theory Y.-T. Siu, Techniques of Extension on Analytic Objects S. R. Caradus et at., Calkin Algebras and Algebras of Operators on Banach Spaces £. O. Roxin et a/., 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 Cesarietal., Nonlinear Functional Analysis and Differential Equations J. J. Schaffer, 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 B. R. McDonald and R. A. Morris, 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 a/., 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 a/., 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 and E. 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 andJ. 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 Stemberg et a/.. 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 a/., 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

60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120.

J. Banas 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 Gee/, 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 a/., 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. 6. Hannsgen et a/., 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-Urruty et al., 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. Istratescu, Strict Convexity and Complex Strict Convexity V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations H. L. Manocha andJ. 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. Lightbourne 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 etal., Transport Theory, Invariant Imbedding, and Integral Equations P. Clement 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

121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183.

P. S. Milojevi$ 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 etal., 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 etal., Mathematical Population Dynamics S. Coen, Geometry and Complex Variables J. A. Goldstein et a/., Differential Equations with Applications in Biology, Physics, and Engineering S. J. Andima et al., General Topology and Applications P Clement et al., 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. JoshiandA. V. Balakrishnan, Mathematical Theory of Control G. Komatsu and Y. Sakane, Complex Geometry /. J. Bate/man, Geometric Analysis and Nonlinear Partial Differential Equations T. Mabuchi and S. Mukai, Einstein Metrics and Yang-Mills Connections L. Fuchs and R. Gobel, 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 etal., 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. Clement and G. Lumer, Evolution Equations, Control Theory, and Biomathematics M. Gyllenberg and L Persson, Analysis, Algebra, and Computers in Mathematical Research W. O. Bray etal., 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. Sjegel, Comparison Methods and Stability Theory J.-P. Zolesio, Boundary Control and Variation M. Kflzeketal., Finite Element Methods G. Da Prato and L Tubaro, Control of Partial Differential Equations E. Ballico, Projective Geometry with Applications M. Costabel et al.. Boundary Value Problems and Integral Equations in Nonsmooth Domains G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations S. Huggett, Twistor Theory H. Coo/cef a/., Continue D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings K. Jarosz, Function Spaces V. Ancona et al., Complex Analysis and Geometry E. Casas, Control of Partial Differential Equations and Applications N. Kalton etal., Interaction Between Functional Analysis, Harmonic Analysis, and Probability Z. Deng et al., Differential Equations and Control Theory P. Marcellini et al. 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. Agliano, Logic and Algebra X. H. Cao et al., 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

184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235. 236. 237.

J. E. Andersen et al., Geometry and Physics P.-J. Cahen et al., Commutative Ring Theory J. A. Goldstein etal., Stochastic Processes and Functional Analysis A. Sort)/, Complexity, Logic, and Recursion Theory G. Da Prato andJ.-P. Zolesio, 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. Schikhof et a/., p-adic Functional Analysis S. Sertoz, Algebraic Geometry G. CaristiandE. Mitidieri, Reaction Diffusion Systems A. V. Fiacco, Mathematical Programming with Data Perturbations M. Krizek et al., Finite Element Methods: Superoonvergence, 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 andL Salce, Abelian Groups, Module Theory, and Topology Z. Chen etal., 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. HudzikandL 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 A. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis R. Salvi, The Navier-Stokes Equations F. U. Coelho and H. A. Merklen, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L Tubaro, Stochastic Partial Differential Equations and Applications W. A. Camiellietal., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes etal., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators G. R. Goldstein et al., Evolution Equations A. Giambruno et al.. Polynomial Identities and Combinatorial Methods A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups J. Bergen et al., Hopf Algebras

Additional Volumes in Preparation

Preface

The ALGEBRA CONFERENCE - VENEZIA 2002 was held at the Venice International University (VTU) on the beautiful island of San Servolo in the Venice lagoon, from June 3 to 8, 2002. More than 190 mathematicians from 35 different countries attended the conference. Twenty-two mathematicians from Eastern Europe, Africa and Asia were supported by the Unesco-Roste and by the organizing committee. The scientific committee consisted of: D. Dikranjan, A. Facchini, M. Fontana, L. Fuchs, R. Gilmer, R. Gobel, K. Fuller, W. Heinzer, C. Ringel and D. Simson. The organizing committee consisted of: F. Barioli, S. Bazzoni, R. Colpi, S. Gabelli, E. Gregorio, C. Metelli, L. Salce, F. Stumbo, A. Tonolo and P. Zanardo. We would like to express our heartfelt appreciation to the members of these committees for their work in making the conference so enjoyable and successful. We also extend our gratitude to-the participants for making the conference so stimulating. The conference was organized in 12 main lectures and 6 plenary lectures in the mornings, and in six different sections in the afternoons, where more than 110 communications were presented. The six sections were as follows: Abelian groups, algebras and their representations, commutative rings, module theory, ring theory, and topological algebraic structures. The content of these Proceedings reflects the choice of these topics. The section on commutative rings was intended to celebrate the 30-year anniversary of the publication of Robert Gilmer's monograph "Multiplicative Ideal Theory". Indeed, Gilmer was one of the main speakers, and seven leading researchers in commutative ring theory gave survey talks on the development of the subject in the last 30 years. The success of the conference was made possible by the support of many institutions, which must be warmly thanked: Ministero dell'Istruzione, Universita e Ricerca (MIUR), Unesco-Roste, Istituto Nazionale di Alta Matematica (INDAM), and the Universities of Milano, Napoli, Padova, Roma 3 and Verona. Also, the "Casino di Venezia" provided substantial support, thanks to the scientific sensibility of its Director, Prof. Armando Favaretto. Special thanks must also be addressed to the Dean of the Venice International University, Prof. Ignazio Musu, and to the

Preface

General Secretary, Dr. Francesca Nisii, for their help and hospitality at San Servolo. The success was completed by exceptional high tides on two evenings; these gave rise to the phenomenon called "acqua alta", that the participants enjoyed very much. A declared goal of the conference was to bring together mathematicians working in different areas of algebra, in order to make evident the connections and interactions among these areas. We believe that these Proceedings reflect the success of the conference in achieving this goal, and we trust that the reader will find confirmation of this. Sheila Brenner was one of the distinguished participants of the Conference. In the official picture, she appears over Sylvia Wiegand and below Chuck Vinsonhaler, under the palm. Sheila passed away on October 10, just a few months after the conferences. She was a prominent figure in the algebraic panorama during the last thirty years. We will miss her.

Alberto Facchini Evan Houston Luigi Salce

Contents

Preface Participants to the Conference Contributors

Hi ix xvii

1.

Additives Galois Theory of Modules R. Abraham and P. Schultz

1

2.

Finitely Generated and Cogenerated QD Groups U. Albrecht and B. Wickless

13

3.

Direct Limits of Modules of Finite Projective Dimension L. Angeleri Hugel and J. Trlifaj

27

4.

Classification of a Class of Almost Completely Decomposable Groups E. Blagoveshchenskaya

45

5.

A Polynomial Ring Sampler J. W. Brewer

55

6.

The Picard Group of the Ring of Integer-valued Polynomials on a Valuation Domain J.-L. Chabert

63

7.

A Note on Cotilting Modules and Generalized Morita Duality R. R. Colby, R. Colpi, andK. R. Fuller

85

8.

Dualities Induced by Cotilting Bimodules R. Colpi

89

vi 9.

Contents Symmetries and Asymmetries for Cotilting Bimodules G. D 'Este

103

10. A Constructive Solution to the Base Change Decomposition Problem inB^-groups C. De Vivo and C. Metelli

119

11.

133

On a Property of the Adele Ring of the Rationals D. Dikranjan and U. Zannier

12. On Strong Going-between, Going-down, and their Universalizations D. E. Dobbs and G. Picavet

139

13. Factorization of Divisorial Ideals in a Generalized Krull Domain S. El Baghdadi

149

14. Divisorial Multiplication Rings J. Escoriza and B. Torrecillas

161

15.

177

Global Deformations of Lie Algebras A. Fialowski

16. Maximal Prime Divisors in Arithmetical Rings L. Fuchs, W. Heinzer, and B. Olberding

189

17. On Strongly Flat Modules over Matlis Domains L. Fuchs, L. Salce, andJ. Trlifaj

205

18. Group Identities on Unit Groups of Group Algebras A. Giambruno and C. Polcino Milies

219

19. Forty Years of Commutative Ring Theory R. Gilmer

229

20.

Modules Induced from a Normal Subgroup of Prime Index S. P. Glasby

257

21.

Uniquely Transitive Torsion-free Abelian Groups R. GobelandS. Shelah

271

22.

Generalized .E-rings R. Gobel, S. Shelah, and L. Strungmann

291

23.

Butler Modules and Bext P. Goeters and C. Vinsonhaler

307

24.

Remarks on the Multiplicative Structure of Certain One-dimensional Integral Domains F. Halter-Koch, W. Hassler, and F. Kainrath

25.

Non-fmitely Generated Prime Ideals in Subrings of Power Series Rings W. Heinzer, C. Rotthaus, andS. Wiegand

321

333

Contents

vii

26. Rings with Finitely Many Orbits under the Regular Action Y. Hirano

343

27.

Projective Covers and Injective Hulls in Abelian Length Categories O. Kerner

349

28.

Semigroup Rings that Are Inside Factorial and Their Cale Representation U. Krause

353

29. Reduced Modules T.-K. Lee and Y.Zhou

365

30. The Mori Property in Rings with Zero Divisors T. G. Lucas

379

31.

401

Minimal Prime Ideals and Generalizations of Factorial Domains P. Malcolmson and F. Okoh

32. Factorizations of Monic Polynomials S. Me Adam and R. G. Swan 33. Monotone Complete Ordered Fields, Generalized Power Series, and Integer Parts M. MoniriandJ. S. Eivazloo

411

425

34. On Distance between Finite Rings L. Oyuntsetseg and T. Sumiyama

431

35. Affine Pairs /. J. Papick

437

36. When a Super-decomposable Pure-injective Module over a Serial Ring Exists G. Puninshi

449

37. Path Coalgebras of Quivers with Relations and a Tame-wild Dichotomy Problem for Coalgebras D. Simson 38. On Central Galois Algebras of a Galois Algebra G. Szeto and Lianyong Xue

465 493

Participants to the Conference

Gene Abrams University of Colorado, USA [email protected]

Ibrahim Assem Universite de Sherbrooke, Quebec, Canada ibrahim.assemSDMI.USherb.CA

Grigori Amosov Moscow Institute of Physics and Technology, Russia gramosOdeom.chph.ras.ru

Giuseppe Baccella Universita di L 'Aquila, Italy baccellafiunivaq.it Ayman Badawi Birzeit University, Palestine abringSbirzeit.edu

Lidia Angeleri Hiigel Ludwig-Maximilians- Universitdt Munchen, Germany

Valentina Barucci Universita La Sapienza, Italy barucciSmat.uniromal.it

lidia.angeleriSuninsubria.it

Pham Ngpc Anh Renyi Institute, Hungarian Academy of Sciences, Hungary anhSrenyi.hu

Silvana Bazzoni Universita di Padova, Italy bazzonitaath.unipd.it

Ramon Antoine Riolobos Universitat Autonoma de Barcelona, Spain ramonOmat.uab.es

Ali Benhissi Faculty of Sciences, Monastir, Tunisia ali-benhissiSyahoo.fr

Pere Ara Universitat Autonoma de Barcelona, Spain paraSmat. uab. es

Ekaterina Blagoveshchenskaya St. Petersburg State Technical University, Russia kateSrobotek.ru

Gonzalo Aranda Pino Universidad de Malaga, Spain gonzaloOagt.cie.uma.es

Prauke Bleher University of Iowa, USA fbleherSmath.uiowa.edu

Nurcan Argac Ege University, Turkey argacOsci.ege.edu.tr David Arnold Baylor University, USA David_Arnold8baylor. edu

Grzegorz Bobiriski Nicholas Copernicus University, Toruri, Poland gregbob8mat.uni.torun.pi

Vyatcheslav Artamonov Moscow State University, Russia artamonSmech.math.msu.su

Bela Bodi University of Debrecen, Hungary bodibelaSmath.kite.hu IX

Participants to the Conference

Sheila Brenner Liverpool University, UK James Brewer Florida Atlantic University, USA brewerOmath.fau.edu Hans Brungs University of Alberta, Canada brungsflualberta.ca Daniel Bulacu Univ. of Bucharest, Romania dbulacu8al.math.unibuc.ro Walter Burgess University of Ottawa, Canada wdbsgQuottawa.ca Michael C. R. Butler Liverpool University, UK mcrbfiliv.ac.uk Paul-Jean Cahen Universite d'Aix-Marseille 3, France paul-jean.cahen8univ.u-3mrs.fr

Septimiu Crivei Babe§-Bolyai University of Cluj-Napoca, Romania criveiSmath.ubbcluj.ro Juan Cuadra University of Almeria, Spain jcdiaz8ual.es Marco D'Anna Universita di Catania, Italy mdanna8dmi.uni ct.it Gabriella D'Este Universita di Milano, Italy Gabriella.DesteSmat.unimi.it Andrew Dean Nipissing University, Canada andrewdSnipissingu.ca Dikran Dikranjan Udine University, Italy dikranj an8dimi.uniud.it Luca Diracca Universita di Padova, Italy dirlucaSlibero.it

Grigore Calugareami Kuwait University, Kuwait caluSmcs.sci.kuniv.edu.kw

Vlastimil Dlab Carleton University, Canada vdlabSmath.carleton.ca

Jean-Luc Chabert Universite de Picardie, Amiens, France j ean-luc.chabertSu-picardie.fr

David Dobbs University of Tennessee, USA dobbsSmath.utk.edu

Scott Chapman Trinity University, USA schapman8trinity.edu

Semra Dogruoz University of Afyon Kocatepe, Turkey dogruoz428hotmail.com

Stephen Chase Cornell University, USA sucl8cornell.edu

Manfred Dugas Baylor University, USA Manfred_Dugas8baylor.edu

Ted Chinburg Univ. of Pennsylvania, ted8math.upenn.edu

USA

Riccardo Colpi Universita di Padova, Italy colpiSmath.unipd.it Pablo Cordero Universidad de Malaga, Spain pcorderoQuma.es Anthony Corner Oxford & Exeter, UK [email protected], c orner8vax.oxf ord.ac.uk

Robert El Bashir Charles University, Prague, Czech Rep. bashir8karlin.mff.cuni.cz Salma Elaoud Faculte des Sciences de Tunis, Tunisia Jose Escoriza-Lopez University of Almeria, Spain jescorizQual.es Sergio Estrada Dominguez Universidad de Almeria, Spain sestradaSual.es Alberto Facchini Universita di Padova, Italy facchiniSmath.unipd.it

Participants to the Conference

Wang Fanggui Nanjing University, China wangrylqflpubli c1.ptt.j s.en

Stephen Glasby Central Washington Univ., USA glasbysScwu.edu

Rolf Farnsteiner University of Bielefeld, Germany RolfFarnSaol.com

Sarah Glaz University of Connecticut, USA glazSuconnvm.uconn.edu

Vitor Ferreira University of Sao Paulo, Brazil vitorflicmc.sc.usp.br

R.GoebelSuni-essen.de

Alice Fialowski Eotvos Lordnd University Budapest, Hungary fialowskOcs.elte.hu

Brendan Goldsmith Dublin Institute of Technology, Ireland brendau.goldsmithSdit.ie

Alev Firat Ege University, Turkey firatSalpha.sci.ege.edu.tr

Miguel Angel Gomez Lozano Universidad de Malaga, Spain

Marco Fontana Universitd 'Roma Tre', Italy fontanafimat.uniromaS.it Manuel Forero Universidad de Cadiz, Spain ajesus.calderonauca.es Laszlo Fuchs Tulane University, USA fuchsStulane.edu Kent Fuller University of Iowa, USA kfullerflmath.uiowa.edu

Riidiger Gobel University of Essen, Germany

Enrico Gregorio Universitd di Verona, Italy gregorioSsci.univr.it Pedro Antonio Guil Asensio University of Murcia, Spain paguilOum.es Ofer Hadas Tel-Aviv Academic College, Israel hadasoSmember. ams.org

Alexander Hahn University of Notre Dame, USA hahn.lSnd.edu

Stefania Gabelli Universita 'Roma Tre', Italy gabelliSmat.uniromaS.it

Franz Halter-Koch University of Graz, Austria franz.halterkochSuni-graz.ac.at

Maria Luisa Galvao University of Lisbon, Portugal mlgalvaoSptmat.lmc.fc.ul.pt

Wolfgang Hassler University of Graz, Austria Wolfgang.hasslerSuni-graz.ac.at

J. Ignacio Garcia-Garci'a Universidad de Granada, Spain jiggSugr.es

Roozbeh Hazrat University of Bielefeld, Germany rhazratSmathemat ik.uni-bielefeld.de

Josefa Maria Garcia Hernandez University of Granada, Spain jgarciahSugr.es

William Heinzer Purdue University, USA heinzer8math.purdue.edu

Alfred Geroldinger University of Graz, Austria alfred.geroldingerSuni-graz. at

Dolors Herbera Universitat Autonoma de Barcelona, Spain dolorsSmat.uab.es

Robert Gilnier Florida State University, USA gilmer8math.fsu.edu

Yasuyuki Hirano Okayama University, Japan yhiranoSmath.okayama-u.ac.jp

Florida Girolami Universita 'Roma Tre', Italy girolamiSmat.uniromaS.it

Sana Hizem Ipest, [email protected]

Participants to the Conference

Evan Houston UNO Charlotte., USA eghoustoSemail.uncc.edu

Piroska Lakatos University of Debrecen, Hungary lapiSmath.kite.hu

Frantjois Huard Bishop's University, Canada fhuardQubishops.ca

David Lantz Colgate University, USA dlantzSmail.Colgate.edu

All Jaballah The University of Sharjah, UAE aj aballahfisharj ah.ac. ae

Zbigniew Leszczynski Nicholas Copernicus University, Torun, Poland zblesSmat.uni.torun.pi

Pascual Jara Univ. Granada, Spain [email protected]

Fang Li Zhejiang University, China f angliamail.hz.zj. en

Byung Gyun Kang POSTECH, South Korea bgkangOpostech.ac.kr Muge Kanuni Massachusetts Maritime Academy, USA mkanuniSmma.mass.edu

Henri Lombardi Universite de Franche-Comte, Besancon, France lombardiOmath.univ-fcomte.fr Alan Loper Ohio State University, USA loperaSmath.Ohio-state.edu

Stanislaw Kasjan Nicholas Copernicus University, Torun, Poland skasj anfimat.uni.torun.pi

Thomas Lucas Univ. North Carolina Charlotte, USA tglucasSemail.uncc.edu

Patrick Keef Whitman College, USA keef8whitman.edu

Adolf Mader University of Hawaii, USA adolfOmath.hawaii.edu

Otto Kerner H.-Heine-Universitdt Diisseldorf, kernerScs.uni-duesseldorf.de

Germany

Abdenacer Makhlouf University Of Haute Alsace, France N.MakhloufSuniv-mulhouse.fr

Toshiko Koyama Ochanomizu University, Japan koyamafiis.ocha.ac.jp

Peter Malcolmson Wayne State University, USA petemSmath.wayne.edu

Henning Krause University of Bielefeld, Germany henningSmathematik.uni-bielefeld.de

Francesca Mantese Universitd di Padova, Italy fmantesefimath.unipd.it Laszlo Marki Renyi Institute, Hungarian Academy of Sciences, Hungary

Ulrich Krause Universitdt Bremen, Germany krausefimath.uni-bremen.de

markifirenyi.hu

Franz-Viktor Kuhlmann University of Saskatchewan, Canada fvkasuoopy.usask.ca

Wallace Martindale Univ. of Massachusetts, USA jmartindSchapline.net

Salma Kuhlmann University of Saskatchewan, Canada skuhlmanSsnoopy.usask.ca

Warren May University of Arizona, USA mayfimath.arizona.edu

Kenneth Kunen University of Wisconsin, USA kunenSmath.wise.edu

Stephen McAdam University of Texas, USA mcadamSmath.utexas.edu

Participants to the Conference

Xlll

Claudia Menini Universita di Ferrara, Italy menSdns.unif e.it

MiHee Park Universita 'Roma Tre', Italy mhparkSeuclid.postech.ac.kr

Claudia Metelli Universita di Napoli, Italy cmetelli8math.unipd.it

Sivasubramaniam Parvathi University of Madras, India sparvathiShotmail.com

Chiara Milan Universita di Firenze, Italy [email protected]

Prancesc Perera Queen's University Belfast, UK pereraSqub.ac.uk

Gigel Militaru University of Bucharest, Romania [email protected]

Gabriel Picavet Universite Blaise Pascal, France [email protected]

Todor Mollov University of Plovdiv, Bulgaria mollovSpu.acad.bg

Martine Picavet Universite Blaise Pascal, France Martine.Picavetamath.univ-bpclermont.fr

Mojtaba Moniri Tarbiat Modarres University, Tehran, Iran

Giampaolo Picozza Universita 'Roma Tre', Italy picozzaSmatrmS.mat,uniroma3,it

mojmonSipm.ir

Nadeem Nadeem-ur-Rehman University Kaiserslautern, Germany rehmanlOOSpostmark.net Georgios Nassopoulos University of Athens, Greece gnassopQmath.uoa. gr Constantin Nastasescu University of Bucharest, Romania cnastaseOal.math.unibuc.ro

Zygmunt Pogorzaly Nicholas Copernicus University, Torun, Poland [email protected] Cesar Polcino Milies Universidade de Sao Paulo, Brazil polcinoSime.usp.br Gena Puninski Manchester University, UK gpuninskiSmaths.man.ac.uk

Maria Teresa Nogueira University of Lisbon, Portugal tnogueirSalf1.cii.fc.ul.pt

Robert Raphael Concordia University, USA raphaelSalcor.concordia.ca

Takashi Okuyama Toba National College of Maritime Technology, Japan okuyamaQmath.hawaii.edu

Idun Reiten Department of Mathematical Sciences, NTNU, Norway idunrSmath.ntnu.no

Bruce Olberding The University of Louisiana at Monroe, USA maolberdingSulm.edu

Paulo Ribenboim Queen's University, Canada mathstatSmast.queensu.ca

Lkhangah Oyuntsetseg Okayama University, Japan oyunaaSmath.okayama-u.ac.jp

Bettina Richmond Western Kentucky University, USA bettina.richmond®wku.edu

Alexander Panov Samara State University, Russia [email protected]

Tom Richmond Western Kentucky University, USA Tom.RichmondSwku.edu

Ira Papick University of Missouri, USA papickiSmissouri.edu

Nicola Rodino Universita di Padova, Italy rodinofimath.unipd.it

Participants to the Conference

Moshe Roitman University of Haifa, Israel mroitmanfimath.haifa.ac.il

William Smith University of North Carolina, USA

Luigi Salce Universita di Padova, Italy salcefimath.unipd.it

Oyvind Solberg Institutt for matematiske fag, Norway

Catarina Santa-Clara University of Lisbon, Portugal cgomesfilmc.fc.ul.pt

Lutz Striingmann Hebrew University of Jerusalem, Israel lutzSmath.huj i.ac. il

Manuel Saon'n Universidad de Murcia, Spain msaorincfium.es

Fabio Stumbo Universita di Ferrara, Italy stffidns.unife.it

Markus Schmidmeier Florida Atlantic University, USA mschmidm8fau.edu

Takao Sumiyama Aichi Institute of Technology, Japan

Phill Schultz The University of Western Australia, Australia schultzSmaths.u«a.edu.au

Jeno Szigeti University of Miskolc, Hungary [email protected]

Chun Seung Ju POSTECH, South Korea bgkangSpostech.ac. kr Bert Sevenhant Limburgs Universitair Centrum, Belgium bert.sevenhantSluc.ac.be Dmitri Shakhmatov Ehime University, Japan dmitriOdpc.ehime-u.ac.jp

wwsmithSemail.unc.edu

oyvinsofimath.ntnu.no

sumiyama8ge.aitech.ac.jp

Earl Taft Rutgers University, USA etaftfimath.rutgers.edu Prancesca Tartarone Universita 'Roma Tre', Italy tfrancefimatrmS.mat.uniromaS.it Noriko Tone Tokyo Denki University, Japan toneficck.dendai.ac.jp

Jay Shapiro George Mason University, USA jshapiroSgmu.edu

Alberto Tonolo Universita di Padova, Italy

Michael Siddoway Colorado College, USA msiddowayficoloradocollege.edu

Sonia Trepode Universidad Nacional de Mar del Plata, Argentina strepodefimdp.edu.ar

Carlos Signoret Universidad Autdnoma Metropolitana, Mexico [email protected], signoretSsmm.org.mx Mercedes Siles Molina Universidad de Malaga, Spain mercedesfiagt.cie.uma.es Daniel Sinason Nicholas Copernicus University, Toruii, Poland simsonSmat.uni.torun.pl Andrzej Skowronski Nicholas Copernicus University, Torun, Poland skowronfimat.uni.t orun.pi

tonolofimath.unipd.it

Jan Trlifaj Charles University, Prague, Czech Rep. trlifajSkarlin.mff.cuni.cz Hisa Tsutsui Millersville University, USA HisafiTsutsui.net Andrzej Tyc Nicholas Copernicus University, Torun, Poland atycSmat.uni.torun.pl Goniil Uslu Ege University, Turkey firatSalpha.sci.ege.edu.tr

Participants to the Conference

Anita Valenta University of Trondheim, Norway valent aSmath.ntnu.no

Larry Xue Bradley University, USA Ixueflhilltop.bradley.edu

Peter Vamos Exeter, UK p.vamosSex.ac.uk

Anatoly Yakovlev St. Petersburg State University, Russia yakovlevSyak.pdmi.ras.ru

Charles Vinsonhaler University of Connecticut, USA vinsonSmath.uconn.edu Filippo Viviani Universitd 'Roma Due', Italy vivianifimat.uniroma2.it Simone Wallutis University of Essen, Germany simone.uallutisOuni-essen.de Bill Wickless University of Connecticut, USA wicklessSmath.uconn.edu Roger Wiegand University of Nebraska, USA rwiegandflmath.unl.edu Sylvia Wiegand University of Nebraska, USA swiegandSmath.unl.edu Richard Wiegandt Renyi Institute, Hungarian Academy of Sciences, Hungary [email protected]•

Paolo Zanardo Universita di Padova, Italy pzanardoSmath.unipd.it Mark Zeldich zeldichSmail.ru Julius Zelmanowitz University of California, USA juliusflmath.ucsb.edu Jan Zemlicka Charles University, Prague, Czech Rep. zemlickaakarlin.mil.cuui.cz Yiqiang Zhou Memorial University of Newfoundland, Australia zhouSmath.mun.c a Grzegorz Zwara Nicholas Copernicus University, Torun, Poland [email protected]

Contributors

Ross ABRAHAM, Department of Mathematics and Statistics, South Dakota State University, Brookings, South Dakota 57007, USA Ross_Abraham8sdstate.edu ULRICH ALBRECHT, Department of Mathematics, Auburn University, Auburn, AL 36849, USA albreuf8math.auburn.edu LlDIA ANGELERI HUGEL, Mathematisches Institut der Universitat, Theresientrasse 39, D-80333 Miinchen, Germany lidia.angelerifiuninsubria.it

EKATERINA BLAGOVESHCHENSKAYA, Department of Mathematics, St. Petersburg State Technical University, Polytechnicheskaya 29, St. Petersburg 195251, Russia kate8robotek.ru, kblag20028yahoo.com J. W. BREWER, Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA [email protected]

JEAN-LUC CHABERT, Department of Mathematics, Universite de Picardie, 80039 Amiens, France, LAMFA CNRS-UMR 6140 j ean-luc.chabertSu-picardie.fr

ROBERT R. COLBY, Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA R. COLPI, Dipartimento di Matematica Pura e Applicata, Universita di Padova, Via Belzoni 7, 35131 Padova, Italy colpiSmath.unipd.it GABRIELLA D'EsTE, Dipartimento di Matematica, Universita di Milano, Via Saldini 50, 20133 Milano, Italy gabriella.desteSraat.unimi.it CLORINDA DE Vivo, Dipartimento di Matematica e Applicazioni, Universita Federico II di Napoli, Italy devivo8matna2.dma.unina.it DIKRAN DIKRANJAN, Dipartimento di Matematica e Informatica, Universita di Udine, Via delle Scienze 206, 33100 Udine, Italy dikranj a8dimi.uniud.it DAVID E. DOBBS, Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300, USA dobbsSmath.utk.edu

XVII

Contributors

JAFAR S. EIVAZLOO, Department of Mathematics, Tarbiat Modarres University, Tehran, Iran eivazloo8ipm.ir, eivazl_j8modares.ac.ir SAID EL BAGHDADI, Department of Mathematics, Faculte des Sciences et Techniques, P.O. Box 523, Beni Mellal, Morocco baghdadiQf stbm.ac.ma JOSE ESCORIZA, Department of Algebra and Mathematical Analysis, University of Almeria, 04120, Almeria, Spain jescoriz8ual.es ALICE FIALOWSKI, Department of Applied Analysis, Eotvos Lorand University, Pazmany Peter setany 1, H-1117 Budapest, Hungary f ialowsMcs. elte. hu LASZLO FUCHS, Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, USA fuchsfitulane.edu KENT R. FULLER, Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA kfullerOmath.uiowa.edu A. GIAMBRUNO, Dipartimento di Matematica, Uiiiversita di Palermo, Via Archirafi 24, 90123 Palermo, Italy a.giambrunoSunipa.it ROBERT GILMER, Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510, USA gilmerSmath.f su.edu S. P. GLASBY, Department of Mathematics, Central Washington University, WA 98926-7424, USA GlasbyS®c«u.edu RUDIGER GOBEL, Fachbereich 6, Mathematik und Informatik, Universitat Essen, 45117 Essen, Germany R.GoebelSUni-Essen.De PAT GOETERS, Department of Mathematics, Auburn University, Auburn, AL 36849-5310, USA FRANZ HALTER-KOCH, Institut fur Mathematik, Karl-Franzens-Universitat Graz, Heinrichstrafie 36, 8010 Graz, Austria franz.halterkochSuni-graz.at WOLFGANG HASSLER, Institut fur Mathematik, Karl-Franzens-Universitat Graz, Heinrichstrafie 36, 8010 Graz, Austria Wolfgang.hasslerSuni-graz.at WILLIAM HEINZER, Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA heinzerfimath.purdue.edu YASUVUKI HIRANO, Department of Mathematics, Okayama University, Okayama 700-8530, Japan yhir anoSmath.okay ama-u.ac.jp FLORIAN KAINRATH, Institut fiir Mathematik, Karl-Franzens-Universitat Graz, HeinrichstraSe 36, 8010 Graz, Austria florian.kainrathfiuni-graz.at OTTO KERNER, Mathematisches Institut, Heinrich-Heine-Universitat, D-40225 Diisseldorf, Germany kernerOcs.uni-duesseldorf.de

Contributors

xix

ULRICH KRAUSE, Department of Mathematics, University of Bremen, Bibliothekstr. 1, Bremen, Germany krauseOmath.uni-bremen.de Tsiu-KwEN LEE, Department of Mathematics, National Taiwan University, Taipei 106, Taiwan tklee8math.ntu.edu.tw THOMAS G. LUCAS, Department of Mathematics, University of North Carolina Charlotte, Charlotte, NC 28223, USA tglucasSemail.uncc.edu PETER MALCOLMSON, Department of Mathematics, Wayne State University, Detroit, Michigan, USA pet emSmath.wayne.edu STEPHEN McADAM, The University of Texas, Department of Mathematics, 1 University Station C1200, Austin TX 78712-0257, USA mcadam8math.utexas.edu CLAUDIA METELLI, Dipartimento di Matematica e Applicazioni, Universita Federico II di Napoli, Italy cmetelliSmath.unipd.it MOJTABA MONIP.!, Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran mojmonfiipm.ir, moniri_m®modares.ac.ir FRANK OKOH, Department of Mathematics, Wayne State University, Detroit, Michigan, USA okohfimath.wayne.edu BRUCE OLBERDING, Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001, USA olberdinSemmy.nmsu.edu LKHANGAA OYUNTSETSEG, Department of Mathematics, Okayama University, Okayama 700-8530, Japan IRA J. PAPICK, University of Missouri-Columbia, Department of Mathematics, Columbia, Missouri 65211, USA GABRIEL PICAVET, Laboratoire de Mathematiques Pures, Universite Blaise Pascal, 63177 Aubiere Cedex, France Gabriel.Picavetamath.univ-bpclermont.fr C. POLCINO MlLIES, Institute de Matematica e Estatistica, Universidade de Sao Paulo, Caixa Postal 66.281, 05315-970 Sao Paulo, Brazil polcinoQime.usp.br GENA PUNINSKI, Department of Mathematics, The Ohio State University at Lima, 435 Galvin Hall, 4240 Campus Drive, Lima Ohio 45804, USA puninskiy.ISosu.edu CHRISTEL ROTTHAUS, Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA L. SALCE, Dipartimento di Matematica Pura e Applicata, Via Belzoni 7, 35131 Padova, Italy salceSmath.unipd.it PHILL SCHULTZ, Department of Mathematics and Statistics, The University of Western Australia, Nedlands, Australia 6907 schultzSmath.uwa.edu.au SAHARON SHELAH, Department of Mathematics, Hebrew University, Jerusalem, Israel, and Rutgers University, New Brunswick, NJ, USA ShelahSmath.huj i.ac. il

Contributors

DANIEL SiMSON, Faculty of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Toruri, Poland simsonSmat.uni.torun.pi LUTZ STRUNGMANN, Fachbereich 6, Mathematik und Informatik, Universitat Essen, 45117 Essen, Germany Iutz.struengmann8uni-essen.de TAKAO SUMIYAMA, Center of General Education, Aichi Institute of Technology, Toyota 470-0392, Japan RICHARD G. SWAN, Department of Mathematics, The University of Chicago, Chicago IL 60637, USA swanSmath.uchicago.edu GEORGE SZETO, Department of Mathematics, Bradley University, Peoria, Illinois 61625, USA szetoflhilltop.bradley.edu BLAS TORRECILLAS, Department of Algebra and Mathematical Analysis, University of Almeria, 04120, Almeria, Spain [email protected] JAN TRLIFAJ, Katedra algebry MFF UK, Sokolovska 83, 186 75 Prague 8, Czech Republic trlifaj9karlin.mff.cuni.cz CHARLES VINSONHALER, Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA BILL WICKLESS, Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA wicklessSmath.uconn.edu SYLVIA WIEGAND, Department of Mathematics and Statistics, University of Nebraska, Lincoln, NE 68588-0323, USA LIANYONG XUE, Department of Mathematics, Bradley University, Peoria, Illinois 61625, USA IxueOhilltop.bradley.edu UMBERTO ZANNIER, Istituto Universitario di Architettura di Venezia, S. Croce, 191, 30135 Venezia, Italy zannierSiuav.it, zannierfidimi.uniud.it YIQIANG ZHOU, Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland A1C 5S7, Canada zhoufimath.mun.ca

Rings, Modules, Algebras, and Abelian Groups

Additive Galois Theory of Modules Ross Abraham Department of Mathematics and Statistics, South Dakota State University, Brookings, South Dakota 57007, U.S.A. RossJVbrahamSsdstate.edu

Phill Schultz Department of Mathematics and Statistics, The University of Western Australia, Nedlands, Australia 6907 [email protected] Abstract. We define Galois connections between the ideals of the ring of endomorphisms of a module and the invariant submodules of that module. As an application, we describe the lattice of annihilator ideals of the endomorphism ring of a separable abelian p-group.

1. Introduction Let M be a left module over a unital ring R and let £ be the endomorphism ring of M. The object of this paper is to construct Galois connections between the lattice of ideals of £ and the lattice of fully invariant submodules of M. Using the consequent isomorphism between the lattices of Galois closed ideals and closed submodules, we are able to deduce structural properties of £ in terms of those of M and vice versa. Thus this paper complements [9] which found Galois connections between the lattice of normal subgroups of a group A of automorphisms of M and the lattice of A-invariant submodules of M. As an application, let G be a separable p-group and let £ be the endomorphism ring of G. We classify the ideals of £ by Hausen matrices, extending the results of [1] which dealt with the case of bounded G. Then we use the Galois connections to find the ideals of £ which annihilate H or G/H for some fully invariant subgroup H, and those which satisfy a double annihilator condition. 1991 Mathematics Subject Classification. Primary: 16S50, Secondary: 20K10, 20K30. Key words and phrases. Galois connection, endomorphism ring, lattice of ideals, invariant submodules.

R. Abraham and P. Schultz

2. Invariant submodules and ideals Denote by H the set of fully invariant submodules of M and by J the set of ideals of £. It is routine to check that both Ti, and I are complete lattices under inclusion. Let H.op denote the opposite lattice of Ti.. For all H e H, let W = Ann£(H) = {/ 6 £ : Hf = 0} and let H* = Ann£(M/H) = {/ e £ : Mf C if}. Once again it is routine to check that H' and H* are ideals of 8. Lemma 2.1. (a) The mapping H H-> H ' ofH into I reverses order and maps joins to meets. (b) The mapping H i—» H* of T-i°p into X reverses order, and maps joins to meets. Proof, (a) If H C K e W, then K' C H'. Since H, K C tf +#, (#+#)' C Conversely, let f & H' n K'. Then for all z + y e If + K, (x + y)f = xf + yf = 0. Thus H'nK' C(H + K)'. (b) Denote the order in Hop by H by 9' : F ^ F' and *' : / ^ T. This pair of functions determines a Galois connection ([2]) between Ti and Z . Similarly, define 9* : U°v -> / and ** : / -* H op by 9* : // ^ E* and ** : / ^> J*. This too is a Galois connection. A submodule H £ Ti. is called closed if H = H" and open if H = H** . Similarly, / & I is closed if / = J" and open if / = /**. Thus H is closed if and only if H = /' for some /, and open if and only if H = I* for some /. Actually, a slightly stronger property holds: Lemma 2.2. Let H e "H. Then H is closed if and only if H = /' for some closed ideal in I. Similarly, I is closed if and only if I = H' for some closed H € H, H is open if and only if H — /* for some open I £ I, and I is open if and only if I ~ H* for some open H € Ti. Proof. Suppose H is closed. Then H = H". Let / = H'. Then / is closed and H — I' . The proofs for the other three cases are similar. D It follows immediately from [2, Chapter 5, Theorem 19] that the closed invariant submodules and closed ideals form complete lattices, and so do the open invariant

Galois Theory of Modules

submodules and open ideals. Let "H be the lattice of closed submodules of Ti. and H° the lattice of open submodules of W op . Let I be the lattice of closed ideals and J° the lattice of open ideals of I. The general theory of Galois Correspondences (see [2, Chapter 5]) states that 9': Ji —» I is a lattice isomorphism with inverse *' and similarly 0* is a lattice isomorphism from H° to 1° with inverse fy*. The duality between ' and * is better illustrated by an alternative definition in terms of ring actions. Each / 6 £ determines an endomorphism of H € Ti. by restriction. Denote the ring of endomorphisms of H which can be extended to £ by £H. Similarly, each / € £ determines an endomorphism / of the factor module M/H by /: m + H i—> mf + H. Denote the ring of endomorphisms of M/H which can be lifted to £ by £M/H.The ring £ acts on the set S of submodules of M by H H-» Hf and the set T of factor modules of M by M/H i-> M / H f . Denote these actions by £s and £-F respectively. Define relations < on S by inclusion and ^ on F by G/H ^ G/K ifK / defined above. Hence the first row and first column are short exact sequences with j and h the TTf

inclusion maps, i the natural epimorphism of H' onto — TTf

rr/

i

composed with the

IT*

natural isomorphism of ——— onto ———, and o the natural epimorphism

R. Abraham and P. Schultz

ET*

TJ*

of H* onto ———- composed with the natural isomorphism of ——— onto ti M -fi

n n a*

H' + H* H' ' Let r be the natural embedding of

rrt I rr*

—— into £H and let s: f\n *-* f +

(H' + H*). Finally, let t: f + H* >-> / and let 9: / i-> / + (/T + H*). These functions make the last row and column exact. D Corollary 2.5. Let H e H. Then (a) /f' n H* is an ideal of 8 with square zero. (b) H'nH* ^ Hom(M/#, IT) as left-R and right-S bimodules. (c) //Hom(M///, //) = 0 anii s and q are both onto, then £ is a subdirect union [6] of£H and £M/H with kernels H' and H*. Proof. Let f,g&H'r\H*. (a) For all x & M, xf e H so i f o = 0. (b) The map which sends / to the induced map /: M/H —> H is an isomorphism. (c) This statement is equivalent to [6, Theorem 6]. D If / G Z, then / is contained in the kernel of the map p: £ —> f 7 and also in the kernel of the map n: £ —* £ M /^ . Hence the action of £ on M induces actions of £/I on I' and on M/1* and we have the following characterizations of closed and open ideals: Theorem 2.6. Let I e I. 1. The following are equivalent: (a) / is closed] (b) J is t/ie kernel of the natural map £ —+ £* \ (c) £/I is faithful on/'. 2. TTie following are equivalent: (a) / is open; (b) / is the kernel of the natural map £ —> £Mll* • (c) f// is faithful on M/I*. Proof. I. (a) =^> (b) Since / is contained in the kernel of p, we must show that if / £ £ annihilates every element of M that is annihilated by / then / € /. This follows from (a). (b) => (c) The action o f f / / on /' is defined by m(f+I) = mf. So if m(f+I) = 0 for all m e / ' then / is in the kernel of p. Hence / e / so the action is faithful. (c) => (a) Let / S £ annihilate every element of /' which is annihilated by every element of /. Since the action is faithful, / € /. Hence / is closed. 2. The proof is clause by clause similar to the proof of 1. D

Galois Theory of Modules

3. The Ideal Connection Consider now the special case in which M — R considered as left JJ-module. Then £ consists of the right multiplications in R, the submodules of M are the left ideals, and the fully invariant submodules are the ideals of R. Thus H — 1 and all the results of Section 2 hold in this context, so in particular, we have a Galois Connection, called the ideal connection, the closure connection between I and T. Note that the interior connection between Xop and X is vacuous, because for any ideal / of R, the left and right annihilators in R of R/I are just / itself. Because Ji = X, the notation of Section 2 is no longer adequate, since there are two possible interpretations of I'. It is therefore necessary to introduce new notations which make sense for arbitrary unital rings R. Let X be the lattice of ideals of a unital ring R. For all / £ X, let 1. rl ={f € R:If = 0}, the right annihilator of / in R, 2. il = {x € R : xl = 0}, the left annihilator of / in R. The left right closure of I is trl and the right left closure is rtl. An ideal I is left right closed if trl = I and right left closed if rtl = I. Two ring theoretic questions immediately arise: which ideals are both left right and right left closed? Which ideals of a non-commutative ring have the property that trl = rtll Faith deals with some related questions in [3, Proposition 19.10 and Theorem 23.25], but only for special classes of rings. We do not deal with these general problems in this paper. Instead, we apply the ideal connection to the case that R = £, the endomorphism ring of a module M. Thus we have the lattice H of fully invariant submodules of M, the lattice X of ideals of £ and three Galois connections to keep track of. We shall need both the notation of Section 2 and that just introduced. The next proposition is an intrinsic characterization of closed and open ideals. Proposition 3.1. Let I e X. Then (a) If I is right left closed, then I is closed. (b) // / is left right closed, then I is open. Proof, (a) We need only prove that if I = rtl then I" C /. Let / e I", g € £/, and m e M. Since gh — Q for all h 6 /, we have mgh = 0 for all h G I, so mg G /'. This shows that Mg C /', hence (Mg)f = 0 which means gf = 0 for any g & tl. Thus / e rtl = I. The proof for (b) is similar. D Lemma 3.2. Let H e H. Then (a) // H is closed then tH' = H". (b) IfH is open then rH* = H'. (c) // H is closed and open, then H' is right left closed and H* is left right closed. Proof. For all H € H, H*H' = 0, so H* C tH' and H' C rH".

R. Abraham and P. Schultz

(a) Suppose H is closed. Let / £ H*. Then there exists a j ;

(b) / 3 ( M ) = O V ( n l A j Proof, (a) It is clear that for all i and j, p^-n^+L^^og.. annihilates G;[pLW], so (rij — niVj + L(i)) V 0 > a(i,j). Conversely, suppose / = (/y) e £ annihilates G[L]. If for some i and j, /y- has height < (rij - n iV j + L(i)) V 0, then there is a cyclic summand (a) of G» such that afij e Gj has height < rv,- — n» + L(i). Hence the exponent of a/y is greater than n, — (HJ - nt + L(i)) = m — L(i), and so pm-L^afa. _^ o f or pm-L(i)a £ G^pL(i)^ & contradiction. (b) Let k = 0 V (n iA j - L ( j ) ) . It is clear that pkE,ij maps G» into Gj[L(j)]. Conversely, if / = (/y) maps G, into G[p L ^^] then / must have height > k. D We can use Proposition 4.11 to find recognition criteria for closed and open ideals. Corollary 4.12. (a) An ideal matrix ( a ( i , j ) ) is the matrix of a closed ideal I if and only if for each row i there is an integer li > 0 such that i>t then 7(1, j) = ^ - nt + i\ (d) if i > t and j < i then 7(1, j) = (HJ — nt +1) V 0. Proof. Assume / is closed and open. If j(c, c) = 0 for all c G [l,s], then by Corollary 4.12 (a), 7 = 0 contrary to our assumption. Thus t exists and we let I = 7(4, t). That each row i < t is zero is a consequence of Corollary 4.12 (a). If j > i > *j then by invoking Corollary 4.12 parts (a), (b), (b), and (a) successively, we get j(i,j) = 7(1, i) = n» — ki and 7(4, i) = nt — ki = j(t, t) = I , so ki = Ui - j(i,j) = nt-i proving (c). If i > t and j < i we apply Corollary 4.12 (a) and then use the fact from (c) that j(i,i] — rii — nt+£to obtain 7(1, j) = (n^—rii+4) VO = (nj-nj + 7(i,i))VO = (nj -m + ni-nt+l)VQ = (HJ - nt + f) V 0. For the converse, Corollary 4.12 (a) is satisfied with ^ = 7(1,1), i G [l,s]. To satisfy (b), we let kj = rij and Uj = j when £ < nt — rij; and if t > nt — Uj, then we set kj = nt — I and Uj = t. LJ We now find similar descriptions for right and left annihilator ideals. Proposition 4.14. Let I = rc(JV,-). G X. Let (l(i,j)} be the matrix of simp!, (see Definition 4.8). Then: (a) rl is simple with matrix ( a ( i , j } ) where a(i,j) = O V (max{n t A i -7(t,i))} - (n; -n iA .,-))te[i,s] (b) £/ 15 simple with matrix (/3(i, j)) where /3(i,j) = 0 V (max {n^At - 70', t)} - (rij - n i A j))te[i,s] Proof. First note that both (a(i, _/)) and (/3(i,j)) satisfy the SHS axioms, and that the maxima in both parts exist. (a) Let J be the simple ideal with matrix ( a ( i , j ) ) . For all / G / and g £ J, and for all t, j £ [l,s], (fg)tj = Z).e[i,s] f u d i j , a convergent sum. For all i e [l,s], /ti maps Gt intop' 1 '( t ' i ) +ni - ntAi Gi, and 5^- maps Gj into p^^+^-^^Gj. Hence /tigij maps Gt into p fc Gj, where k = ^(t, i)+m — n tAi + a(i, j) + n^ - n iA j. Since a(i, j) > n iAt - j(t,i)) - Ui + n iAj -, fc > nj, so each /t^- = 0. Hence fg = 0so jCrl. Conversely, let g e rl, so each p^ e (rl)ij. Define / as follows: let t G [l,s] and let (a) be a cyclic summand of Gt and (6) a cyclic summand of Gj. Let / map a to pi^^+^-^^b and a complement of (a) to zero. Then / G /, so (fd)tj = ftidij = 0- It follows that gij maps pi^^+^-^^b to zero, so ^j has height > n tAi - l(t,i) - (HJ - nlA.,) = a(i,j). Thus r/ C J. The proof of (b) is similar. D We could now successively apply parts (a) then (b) of Proposition 4.14 to simp I to develop a characterization of left right closed ideals or apply (b) then (a) to characterize right left closed ideals. However, this would yield a formula much more complicated and difficult to use than directly applying Corollary 4.4 and

R. Abraham and P. Schultz

12

Corollary 4.12. Therefore we will omit the formula, but demonstrate the technique in the following example. The example will also serve to illustrate Proposition 4.14 and Corollary 4.12 and support Corollary 4.4. Example 4.15. Let G ^ Z(p3) © Z(p4) © Z(p6). Let I be the ideal of £ with matrix

"1 1

It is routine to check that J satisfies the SHS axioms and Corollary 4.12 (a), so / is closed and hence right left closed, and equals G[L]' with L(i) = i for i = 0,1,2,3. But / fails to satisfy Corollary 4.12 (b) so / is not open or left right closed. Using Proposition 4.14 we calculate the matrices a of rI and /3 of (.1 as

I 2 2 Then the matrices 6 for rtl and e for .

"l

1 -1" 2 2 and (e(i,j)) = 1 3

0 1 1

confirming that I is right left but not left right closed.

References [1] M. A. A vino Diaz and P. Schultz, The endomorphism ring of a bounded abelian p-group, in Abelian Groups, Rings and Modules, Amer. Math. Soc. Series Contemporary Mathematics, 273, (2001), 75-84. [2] G. Birkhoff, Lattice Theory. Amer. Math. Soc. Colloquium Publ. Vol 25 (1967). [3] C. Faith, Algebra II, Ring Theory, Springer-Verlag Series Die Grundlehren der mathematischen Wissenschaften, Band 191 (1973). Vol I (1970). [4] L. Fuchs, Infinite Abelian Groups, Academic Press Vol I (1970). [5] L. Fuchs, Infinite Abelian Groups, Academic Press Vol II (1973). [6] L. Fuchs, On Subdirect Unions Acta Math. Acad. Sci. Hung., 3 (1952) 103-119. [7] J. D. Moore and E. J. Hewitt, On fully invariant subgroups of abelian groups Comment. Math. Univ. St. Pauli, 20 (1971) 97-106. [8] P. Schultz. When is an abelian group determined by the Jacobson radical of its endomorphism ringl, in Abelian Group Theory and Related Topics , Amer. Math. Soc. Series Contemporary Mathematics, 171, (1994), 385-396. [9] P. Schultz, Groups acting on Modules, in Abelian Groups and Modules, Birkhauser Verlag Series Trends in Mathematics, (1999) 75-85.

Finitely Generated and Cogenerated QD Groups Ulrich Albrecht Department of Mathematics, Auburn University, Auburn, AL 36849, USA albreufOmath.auburn.edu Bill Wickless Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA wickless8math.uconn.edu Abstract. A group A is quotient divisible (qd) if A contains no Z(p°°) and there is a finite rank free subgroup F < A with A/F torsion divisible. Motivated by the defining conditions of torsion-free finite rank Butler groups, we study qd groups H, G which can be fit into various kinds of exact sequences: 0 — > H -^ A —> G —> Q. Here A = 0™=1 Ai, where the Ai are specified qd groups, usually of rank one.

1. Introduction All groups will be additive abelian groups. Standard categorical and abelian group theoretic notions and notation, such as found in [F], will be employed. Definition 1. A group G is quotient divisible (qd) if G contains no group of type Z(p°°) and if G contains a finite rank free subgroup F such that G/F is a divisible torsion group. The class T> of quotient divisible groups was introduced in [FWl]. There the authors discussed some properties of T> and the associated category QD with objects groups in D and maps quasi-homomorphisms: HomQx>(X, Y) = Q®z~Rom.z(X, Y). Note that the torsion-free groups in V are just the classical torsion-free finite rank qd groups introduced and studied by Beaumont and Pierce in 1961 [BP]. We record for later reference one key result from [FWl]. Theorem 2. There is an exact rank-preserving duality d between the category QV and the category QTT of torsion-free, finite rank (tffr) abelian groups and quasihomomorphisms. 13

14

U. Albrecht and B. Wickless

It would be useful to be more precise on exactly what is shown in [FW1] . There are rank-preserving contravariant additive functors a' : QT> —> QTJ-, d' : QTf —» QD such that, for each qd A, (d'd)A « A and, for each tffr G, (dd')G ~ G. Here ~ denotes quasi-isomorphism. Abelian groups X, Y are quasi-isomorphic if there are homomorphisms f : X —» Y", g : Y —> X whose composites are each equal to a positive integer times the appropriate identity map. Additionally, if 0 —> H -^> C —» G —> 0 is an exact sequence with objects qd groups, giving rise to the sequence 0 —> dG —> dC —^-> dH —> 0 then we can replace the torsion-free finite rank groups dG, dC, dH, each with a quasi-isomorphic copy of itself (dG)i, (dC)i, (dH)i such that the sequence 0 -» (dG)i -^ (dC)i ^ (d#)i -> 0 is an exact sequence of abelian groups. (Not just an exact sequence in the category Q.TJ-.) A similar claim holds for the functor d'. The above theorem implies that the qd and the tffr groups are the same at the "quasi" level; in particular QD is a Krull-Schmidt category. The indecomposables in QD are the strongly essentially indecomposables, those groups G such that, if G > A © B > nG for some positive integer n, then either A or B must be finite. Nonetheless, there are many situations in which the class D differs significantly from the class of tffr groups. For instance, if A, B are tffr with isomorphic endomorphism rings, there is, except in a few simple cases, no relationship between A and B. In contrast, [PiW] displays a reasonably large class C of qd groups for which the Baer-Kaplansky Theorem holds: If A,B &C and E(A) ^ E(B) then A 3* B. For another example, O'Meara and Vinsonhaler have recently introduced and studied the notion of multi-isomorphism for tffr groups ([O'MVl, O'MV2, O'MVS]). Groups A, B are multi-is omorphic if An = Bn for all natural numbers n > 2. In [W] it is shown that in the class T> the relation of multi-isomorphism has some similar but some quite different characteristics. The goal of this paper is to examine some notions for T> motivated by the definition of the class of Butler groups as a subclass of the class of tffr groups.

2. QD groups, the class T> By the rank of a mixed group G we mean its torsion-free rank; that is the rank of G/T(G). A subgroup F < G is called full if G/F is torsion, equivalently if rankF = rankG. The following facts follow directly from the qd definition: (1) All qd groups must have finite positive rank. (2) If G is qd and H is an epimorphic image of G containing no Z(p°°), then H is qd. Let P be the set of all positive integral primes and let G be a group such that its first Ulm subgroup G1 = r\PzpPuG = (0). Then we can embed G as a pure subgroup of G", the completion of G in its Z-adic topology, via the isomorphism The following theorem both characterizes the reduced groups in T> and has some useful corollaries. The proof, as given in [FW1], is fairly straightforward. Theorem 3 ([FW1]). Let G be a reduced group of finite positive rank. Then G e T> if and only if G1 = (0) and there exists a finite rank full free subgroup

Finitely Generated and Cogenerated QD Groups

15

F < G such that, when G is embedded as a pure subgroup of Y[p^p(G/pu'GYp, G satisfies the projection condition. That is, the projection of F onto each factor (each p-adic completion of G/puG] generates that factor as a Z^-module. Hence, a typical group in T> will be of the form Qi © G, where j > 0 and G satisfies the conditions of Theorem 3, above. Corollary 4. Let G be qd and let q be a prime. Then G = Tq © Gq, where Tq = Tq(G) is finite and Gq has no elements of order q. Proof. Without loss, assume that G is reduced. Then, by the projection condition, each (G/p^GYp is a finitely generated Zp-modu\e. Thus, we can write each (G/p^GYp as a direct sum (G/p"G}"p = Tp ® F(p), where Tp is a finite p-group and F(p) is a finite rank free Zp-module. In this situation Tp will coincide with TP(G), the p-torsion subgroup of G. For each fixed prime q, define Gq = G n [F(q) ® Ylp^G/p^GYp]- It is not hard to check that Gq has no elements of order q and that G = Tq ® G9'. D The following result is a straightforward consequence of the definition. Theorem 5. Let G, H be reduced qd with H < G. If G/H is torsion and reduced, then G/H is finite. Proof. Let F < G, F' < H be finite rank full free subgroups such that G/F, H/F1 are torsion divisible. Since G/H is torsion, rankF = rank F'. It follows that Fr\F' is of finite index in both F and F'. Hence, both G/[F n F'} and H/[F D F'} are finite plus divisible. Since G/H is reduced, it must be that G/H is finite. D Another class of mixed groups, the self-small mixed groups G of finite rank such that G/T(G) = Qn, has been extensively studied by a number of authors. This class, denoted by Q, turns out to coincide with the groups of the form Q}: (B B © GO, where j > 0, B is finite and GO can be embedded as a pure subgroup of the direct product of its p-torsion subgroups [Go < EL^pC^o)] such that GO satisfies the projection condition. (See [AGW], Corollary 2.4 for details.) We close this section by relating the class Q to the class T>. For groups in Q of the form GO above and for each prime q, we have qwGo = Go n Ylp^gTp(G0). Hence, G0/qwG0 ^ Tq(G0) for each q. Furthermore, by the projection condition, each T g (Go) is finite, hence complete. It follows that we can identify Tq(Go) with (Go/g^Go), for each q. With this identification, we see that the class V contains not only the classical Beaumont-Pierce torsion-free finite rank qd groups but also, modulo finite summands, the class Q. We note for future reference that, for a group GO € Q as above, since T(Go) = 0 P T P (G 0 ) < Go < HPTP(G0), then [G0/T(G0)] < n p r p (G 0 )/0 p T p (G 0 ). It follows that G 0 /T(G 0 ) is divisible. Finally, we remark that, for such groups G0 £ Q, the group GqQ of Corollary 4 must be the unique maximal g-divisible subgroup of G 0 . Thus, for G0 £ Q, the complementing summand G' in Corollary 4 is unique; GQ = GO n Ylp^gTp(Go

16

U. Albrecht and B. Wickless

However, it is not hard to construct an example of a group G G T> \ Q such that one of its p-torsion subgroups Tp(G) does not have a unique complementary summand. 3. Finitely ZVgenerated and cogenerated groups It will be convenient to slightly enlarge the class D by allowing finite groups to be considered as trivial qd groups. (Then it will be exactly true that Q C T>.) So a typical element of our new D will be of the form QJ © G ® B, where j > 0, G is a reduced group satisfying the conditions of Theorem 3, and B is finite. A qd group of positive rank will be called nontrivial. Let T>i denote the class of all qd groups of rank one. A little thought shows that the classes defined in (ii) below are the qd analogues of the classes originally defined by Butler in [B]. Definition 6. (i) Let {At : I < i < n} C Vv. Then 0™=1 Ai is called a completely decomposable qd group, (ii) Suppose that H, G are qd and there is an exact sequence 0 —> H —> 0™=1 Ai —> G —> 0 with 0™=1 Ai completely decomposable qd. Then we call G a finitely 'Di-generated group and H a finitely T>-L-cogenerated group. The requirement that H, G be qd is playing the role here played by purity in the definition of Butler groups. Let P be the class of finitely Degenerated groups and TL the class of finitely D!-cogenerated groups. As in the classical case for Butler groups ([B]), we have the following basic result. First, we introduce one more notion that will be convenient both here and later. Definition 7. A sequence 0 — > H -^ C —-> G -^ 0 of torsion-free or qd groups is called quasi-exact (resp. quasi-pure exact) if there are groups HI « H, C\ ~ C, GI ~ G such that 0 —> HI —^ C\ —^-> GI —+ 0 is exact (resp. pure exact). In the torsion-free case, the maps ai, f3i are induced by the maps a, j3 by constructing H I , Ci, GI as subgroups of the divisible hulls QH, QC, QG. For qd groups we first need to discard an n-bounded finite summand from each group X and then construct Xi as a subgroup of Z [ l / n ] ($> X. Theorem 8. The class P coincides with the class Ii. Proof. Let H e P. Then H fits into an exact sequence of the form (f): 0 —» ff

a

0™=1 AJ —> G —> 0 as in Definition 6. Applying the duality d of Theorem 2 produces a quasi-pure exact sequence of tffr groups d(\): Q —*• dG —> 0™=1 dAi —> dff —> 0. Since d is rank preserving, dH" is a Butler group. Hence, there is a pure exact sequence 0 —> dH —+ 0^x .Xi —> L —> 0 with the Xi subgroups of Q. Apply the a

inverse duality d' to obtain a quasi-exact sequence of qd groups: 0 —> cf I/ >5 = 0™! cf'Xj -^+ (d'd}H -+ 0. A short calculation now shows that that (d'd)H G 72.. But d'd = nlfl- for some n > 0. So we have nH € 72. and, by Theorem 5, H/nH is finite. Hence, H £U.

Finitely Generated and Cogenerated QD Groups

17

The proof that 71 C P is very similar, so is left to the reader. In this direction it's convenient to note that, if A is rank one qd and B is finite, then A®B qualifies as a rank one qd group. D It seems reasonable to call a group in the class (P = 1Z) a mixed Butler group. The proof of Theorem 8 leads to the following corollary. Corollary 9. Let 0 —> H -^ An —> G —> 0 be an exact sequence of nontrivial qd groups with A of rank one. Then both H and G are quasi-isomorphic to a direct sum of copies of A. We recall a notion from [FW2]. Let G < Tlp£P M(p] be qd, where M(p) = (G/puG)p. For each prime