Advances in Ring Theory PDF
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Advances in Ring Theory S. K. JAIN S. TARIQ RIZVI Editors Springer Science+Business Media, LLC S. K. Jain S. Tariq Rizvi Department of Mathematics Department of Mathematics Ohio University Ohio State University at Lima Athens, OH 45701 Lima, OH 45804 Library of Congress Cataloging-in-Publication Data Advances in ring theory / S. K. Jain and S. Tariq Rizvi, editors, p. cm. - (Trends in Mathematics) Includes bibliographical references and index. ISBN 978-1-4612-7364-6 ISBN 978-1-4612-1978-1 (eBook) DOI 10.1007/978-1-4612-1978-1 1. Rings (Algebra) I. Jain, S. K. (Surender Kumar), 1938- II. Rizvi, S. Tariq. III. Series. QA247.A28 1997 97-26397 512\4~dc21 CIP. AMS Classification:s 05A10.13C05,13F20,16A08,16A33,16D, 1106D50,16D70, 16D90,16E50,16E60,16L30,16U20,16P40,16N40,16P50,16P60,16P999 ,16S9 Printed on acid-free paper © Springer Science+Busines sMedia New York 1997 )£K ® Originally published by Birkhäuser Boston in 1997 ^ Softcover reprint of the hardcover 1st edition 1997 Copyright is not claimed for works of U.S. Governmetn employee.s All rights reserve.d No part of this publication may be reproduce,d stored in a retrieval system, or transmitted, in any form or by any mean,s electronic, mechanica, lphotocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Springer Science+Busines sMedia, LLC for libraries and other usesr registered with the Copyright Clearance Center (CCC), provided that the base fee of$6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requesst shoudl be addressde directly to Springer Science+Business Media, LLC ISBN 978-1-4612-7364-6 Typeset in TEX by the authosr 9 8 7 6 5 4 3 21 CONTENTS Kasch Modules T. Albu and R. Wisbauer 1 Compactness in Categories and Interpretations P. N. Anh and R. Wiegandt . . . . . . . . . . . . . 17 A Ring ofMorita Context in Which Each Right Ideal is Weakly Self-injective S. Barthwal, S. K. Jain, S. Jhingan, S. R. Lopez-Permouth 31 Splitting Theorems and a Problem of Muller G. F. Birkenmeier, J. Y. Kim and J. K. Park .. . . . . . . 39 Decompositions ofD1 Modules R. A. Brown and M. H. Wright 49 Right Cones in Groups H. H. Brungs and G. Tomer . . . . . . . . . . . . . . . . 65 On Extensions of Regular Rings ofFinite Index by Central Elements W. D. Burgess and R. M. Raphael 73 Intersections ofModules J. Dauns 87 Minimal Cogenerators Over Osofsky and Camillo Rings C. Faith . 105 Uniform Modules Over Goldie Prime Serial Rings F. Guerriero . . . . . . . . . . . . . . . . 119 Co-Versus Contravariant Finiteness ofCategories ofRepresentations B. Huisgen-Zimmermann and S. O. Smal¢ 129 Monomials and the Lexicographic Order H. Hulett . 145 Rings Over Which Direct Sums ofCS Modules Are CS D. V. Huynh and B. J. Muller .. . . . . . . 151 Exchange Properties and the Total F. Kasch and W. Schneider 161 Local Bijective Gabriel Correspondence and Torsion Theoretic FBN Rings P. Kim and G. Kmuse . 175 Normalizing Extensions and the Second Layer Condition K. A. Kasler . . . . . . . . . . . . . . . . . . . 191 vi CONTENTS Generators ofSubgroups ofFinite Index in GLmClG) G. T. Lee and S. K. Sehgal . 211 Weak Relative Injective M-Subgenerated Modules S. Malik and N. Vanaja . . . . . . . . . . . . . . . .. 221 Direct Product and Power Series Formations Over 2-Primal Rings G. Marks 239 Localization in Noetherian Rings M. McConnell and F. L. Sandomierski . 247 Projective Dimension ofIdeals in Von Neumann Regular Rings B. L. Gsofsky . 263 Homological Properties ofColor Lie Superalgebras K. L. Price .. 287 Indecomposable Modules Over Artinian Right Serial Rings S. Singh . 295 Nonsingular Extending Modules P. F. Smith . 305 Right Hereditary, Right Perfect Rings Are Semiprimary M. L. Teply . 313 On the Endomorphism Ring ofa Discrete Module: A Theorem ofF. Kasch J. M. Zelmanowitz . 317 Nonsingular Rings with Finite Type Dimension Y. Zhou . 323 PREFACE This volume is an outcome of invited lectures delivered at the Ring Theory Section of the 23rd Ohio State-Denison Conference in May 1996. It also contains articles by some invited mathematicians who could not attend the conference. These peer-refereed articles showcase the latest developments and trends in classical Ring Theory, highlighting the cross fertilization ofnew techniques and ideas with the existing ones. Providing a wide variety of methodologies, this volume should be valuable both to graduate students as well as to specialists in Ring Theory. We would like to thank our colleagues who invested a lot oftheir time to make the conference a great success. In particular, our thanks go to Professors Tom Dowling, Dan Sanders, Surinder Sehgal, Ron Solomon and Sergio R. L6pez-Permouth for their help. The financial support for the Conference, provided by the Department ofMathematics, The Ohio State University, and Mathematics Research Institute, Columbus, is gratefully acknowleged. Many thanks go to Dean Violet I. Meek for her commitment to the promotion of research by her continuous encouragement of such efforts and for providing financial support from the Lima campus of The Ohio State University. We have received immense cooperation from all the referees who, meticulously and in a very short time, provided us with their reports in spite of their busy schedules. We express our sincere thanks to all of them. Finally, we thank Ms. Cindy White for her excellent job in typing parts of this volume. We are pleased to dedicate this volume to Professor Bruno J. Miiller on the occasion ofhis retirement for his many contributions to the Theory of Rings and Modules. As this volume was going to press we have learned that Professor Carl Faith is retiring this year. It is our great pleasure to dedicate this volumealsoto Carlonhis retirement and onhis 70th birthday for his outstanding works in the field ofAbstract Algebra. S. K. Jain and S. Tariq Rizvi, editors March. 1997 KASCH MODULES TOMA ALBU AND ROBERT WISBAUER Abstract An associative ring R is a left Kasch ring ifit contains a copy of every simple left R-module. Transferring this notion to modules we call aleft R-module M a Kasch moduleifitcontainsacopyofevery simple modulein O"[M]. Theaim ofthis paper is to characterizeand investigate this class ofmodules. INTRODUCTION Let M be a left R-moduleover an associative unital ring R, and denote by aIM] the full subcategory of R-Mod consisting of all M-subgenerated R-modules. In section 1 we collect some basic facts about aIM], torsion theories, and modules of quotients in arM]. In section 2 we introduce the ~ncept of a Kasch module. M is a Kasch module it its M-injective hull M is an (injective) cogenerator in arM]. For RM = RR we regain the classical concept ofleft Kasch ring. Various characterizations ofKasch modules are provided. In section 3 we present some properties ofKasch modules. Note that the notion ofKasch module in [10] and [16] is different from ours. Also the notionofKasch ring used in these papers (R is a Kasch ring ifRR and RRare injectivecogenerators in Mod-R and R-Mod respectively) is different from the usual one. 1 PRELIMINARIES Throughout this paper R willdenotean associative ring with nonzero iden tity, R-Mod the category of all unital left R-modules and M a fixed left R-module. The notation RN will be used to emphasize that N is a left R module. Module morphisms will be written as acting on the side opposite to scalar multiplication. All other maps will be written as acting on the left. S. K. Jain et al. (eds.), Advances in Ring Theory © Springer Science+Business Media New York 1997 2 TOMA ALBU AND ROBERT WISBAUER Any unexplained terminology or notation can be found in [7]' [13], [14] and [15]. 1.1 M-(co-)generated modules. A left R-module X is said to be M generated (resp. M-cogenerated) ifthereexists aset I and anepimorphism M(I) ----+ X (resp. a monomorphism X ----+ MI). The full subcategory of R-Mod consisting ofall M-generated (resp. M-cogenerated) R-modules is denoted by Gen(M) (resp. Cog(M». 1.2 The category arM]. A left R-module X is called M-subgenerated if X is isomorphic to a submodule of an M-generated module, and the full subcategory of R-Mod consisting of all M-subgenerated R-modules is denoted by arM]. This is a Grothendieck category (see [14]) and it determines a filter of left ideals PM = {I ~ RRIR/I E arM] }, which is preciselytheset ofall open left idealsofR in the so called M-adic topology on R (see [6]). For any X E arM] weshall denote byXthe injective hullofX in arM], called also the M-injective hull ofX. With this terminology, the injective hull ofX in R-Mod is the R-injective hull, denoted in the sequel by E(X). It is known (see e.g. [14, 17.9]) that X=Tr(M,X) =Tr(a[M], X), where Tr(M,X) (resp. Tr(a[M], X» denotes the trace of M (resp. arM]) in X. 1.3 Hereditary torsion theories in arM]. The concept of a torsion theorycanbedefined inanyGrothendieckcategory(cf. [8]),soinparticular = in arM]. A hereditary torsion theory in arM] is a pair r (7,F) of nonempty classes of modules in arM] such that 7 is a hereditary torsion classor a localizing subcategoryofarM] (this means that it is closed under subobjects, factor objects, extensions, and direct sums) and F = {X E a[M]IHomR(T,X) = 0, \ITE 7}. The objects in 7 are called r-torsion modules, and the object in Fare called r-torsionfree modules. For any X E arM] we denote by r(X) the r-torsion submodule of X, which is the sum ofall submodules ofX belonging to T. Clearly, one has X E T ¢:} r(X) = X, and X E F ¢:} r(X) = O. KASCH MODULES 3 Note that any hereditary torsion theory T = (T,F) in O"[M] if com pletely determined by its first component T, and so usually the hereditary torsion theories are identified with hereditary torsion classes. Any injective object Q E O"[M], i.e., any M-injective module belonging to dM], determines a hereditary torsion theory 7Q = (TQ,Fq), called the hereditary torsion theory in O"[M] cogenerated by Q: TQ = {X E O"[M] IHomR(X,Q) = O} and FQ = Cog(Q) nO"[M]. Note that for any N E O"[M], Cog(N) n O"[M] is precisely the class CogM(N) ofallobjectsinO"[M] whicharecogeneratedbyN inthecategory O"[M] (i.e., are embeddable in direct products in O"[M] ofcopies of N). According to [15, 9.4, 9.5]' any hereditary torsion theory 7 = (T,F) in O"[M] has this form, i.e., for any such 7 there exists an M-injective module Q in O"[M] with 7 = 7Q. For any M-injective module Qin O"[M] we can also consider the heredi tarytorsion theory7E(Q) = (TE(Q),FE(Q)) in R-Mod cogeneratedby E(Q): TE(Q) = {RX IHomR(X, E(Q)) = O} and FE(Q) = Cog(E(Q)). Since for any X E O"[M] and f E HomR(X,E(Q)), one has Im(J) E Q Tr(O"[M],E(Q)) = = Q, we deduce that TQ = TE(Q) nO"[M] and FQ = FE(Q) nO"[M] , that is, any hereditary torsion theory 7 = (T,F) in O"[M] is the "trace" 7' nO"[M] of a certain hereditary torsion theory 7' = (T',P) in R-Mod: this means that T = T' nO"[M] and F = F' nO"[M]. M 1.4 The Lambek torsion theory in O"[M]. The M-injective hull of the module RM cogenerates a hereditary torsion theory tii = (Tii' Fif) in 0"[M], namely: Tii = {X EO"[M] IHomR(X,M) = O}, Fii = CogM(M) = O"[M] nCog(M), called the Lambek torsion theory in O"[M]. Note that this torsion theory depends on the choice of the subgenerator of O"[M]. If O"[M] = O"[N] for some RN, then in general tii =F TN· 4 TOMA ALBU AND ROBERT WISBAUER If RM = RR then weobtain the torsion theoryTE(R) on R-Mod, which is precisely the well-known Lambek torsion theory in R-Mod. The corre sponding Gabriel topology on R is the set ofall dense left ideals of R. In the sequel, we shall denote by DM the Gabriel topology on R corre spondingto the hereditary torsion theory in R-Mod cogenerated by E(M), 1.5 Modules of quotients in O"[M]. Let T = (T,F) be a hereditary torsion theory in O"[M]. For any module X E O"[M] one defines the T injective hull of ~ (see [15, 9.10]) as being the submodule E-r(X) of the M-injective hull X of X for which The module of quotients Qr(X) of X with respect to T is defined (see [15, 9.14]) by In particularonecanconsiderfor any X E O"[M] the moduleofquotients ofX with respect to the Lambek torsion theory TjJ in O"[M]. The module of quotients of a module RX with respect to the Lambek torsion theory TE(R) in R-Mod is denoted by Qrnax(X) and is called the maximal module of quotients of X. For RX = RR one obtains a ring denoted by Q~ax(R) and called the maximal left ring ofquotients of R. 2 DEFINITION AND CHARACTERIZATIONS The following result is well-known (see e.g. [13, Lemma 5.1, p. 235]): 2.1 Proposition. The following assertions are equivalent for a ring R: (1) DR ={R}; (2) E(R) is an injective cogenerator ofR-Mod; (3) Every simple left R-module is isomorphic to a (minimal) left ideal ofR; (4) HomR(C,R) :F 0 for every nonzero cyclic left R-module C; (5) f(I) :F 0 for every left ideal I ofR, where f(I) ={r E RIrI =O}.
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