Table Of ContentAdvances 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
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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}.
