Module Theory: Papers and Problems from The Special Session Sponsored by The American Mathematical Society at The University of Washington Proceedings, Seattle, August 15–18, 1977

243 Pages·1979·2.26 MB·English

by  J. T. Stafford (auth.), Carl Faith, Sylvia Wiegand (eds.)

Lecture Notes ni
Mathematics
Edited yb .A Dold dna .B nnamkcE
700
Module Theory
srepaP dna Problems from
ehT Special Session Sponsored yb
ehT American Mathematical Society ta
ehT University of Washington
,sgnideecorP Seattle, August 15-18, 7791
Edited yb Carl Faith dna Sylvia Wiegand
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Berlin Heidelberg New kroY
1979
Editors
Carl Faith
Department of Mathematics
Rutgers,
The State University
New Brunswick, New Jersey 08903/USA
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AMS Subject Classifications (1970): 13A15,13 B 20,13 B99,13 C10,13 C15,
13D05, 13F05, 13F15, 13F20, 13H10, 16A08, 16A34, 16A52, 16A62,
16A64, 16A66, 18G XX.
ISBN 3-540-09107-6 Springer-Verlag Berlin Heidelberg New¥ork
ISBN 0-387-09107-6 Springer-Verlag NewYork Heidelberg Berlin
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to fee the of amount the publisher, eb determined yb publisher. the with agreement
© yb Berlin Springer-Verlag Heidelberg 9791
Printed ni Germany
Printing dna binding: Offsetdruck, Beltz Hemsbach/Bergstr.
012345-0413/1412
I" am going ot hang up the gloves next year"
(Nathan Jacobson)
DEDICA TION
The contributors and particpants of the Special Session
dedicate this volume ot Nathan Jacobson ni admiration ,
and gratitude for showing us what ot do in the ring
(theory).
PREFACE
The editors wish to thank the participants and contributors
for their splendid cooperation, and for their joe de vvre which
made the Special Session so much fun.
The senior editor has grouped the contributed papers along
ideological lines whenever possible, although, like much ideology,
these are far-fetched in many cases. tI therefore would serve no
useful purpose ot expose this here, but the reading si certainly
better served this way than the old ABC way. One ought to mention
that Dr. J.T. Stafford appeared first ni the program, as a guest
of the Society introduced by the senior editor, and that Dr. Warfeld' s
paper just has to follow Stafford' s . Et cetera.
LIST OF PARTICIPANTS AND CONTRIBUTORS
G. Azumaya .S Mohamed
Indiana University Kuwait University
Bloomington, IN 47401 Kuwait
J. Beaehy .B .L Osofsky
Northern Illinois Univ. Rutgers University
DeKalb, IL 60115 New Brunswick, NJ 08903
A. .K Boyle .Z Papp
University of Wisconsin George Mason University
Milwaukee, WI 53706 Fairfax, VA 22030
.V Camillo X. Smith
University of Iowa University of Texas
lowa City, IA 52240 Austin, TX 78712
C. Faith J. T. Stafford
Rutgers University Brandeis University
New Brunswick, NJ 08903 Waltham, HA 02154
K. R. Fuller M. .L Teply
University of lowa University of Florida
Iowa City, IA 52240 Gainesville, FL 32611
F. Hansen R. Warfield, Jr.
Universit~t Bochum University of Washington
D - 4630 Boehum Seattle, WA 98195
M. Hochster R. Wiegand
University of Michigan University of Nebraska
Ann Arbor, MI 48109 Lincoln, NB 68588
L. S. Levy .S Wiegand
University of Wisconsin University of Nebraska
Madison, WI 53706 Lincoln, NB 68588
E. Matlis
Northwestern University
Evanston, IL 60201
TABLE OF CONTENTS
Papers
J. Toby STAFFORD: Cancellation for Nonprojective Modules ..... 3
Robert B. WARFIELD, Jr.: Stable Generation of Modules ........ 16
Goro AZUMAYA: Some Aspects of Fuller's Theorem ............... 34
John A BEACHY: On Inversive Localization ..................... 46
Ann K. BOYLE and
Edmund H. FELLER : Semicritical Modules and k-Primitive Rings. 57
K.R. FULLER and. A Note on Loewy Rings and Chain Conditions
V.P. CAMILLO on Primitive Ideals ........................ 75
Saad MOHAMED and
Bruno J. FfOLLER : Decompositions of Dual-Continuous Modules .. 87
Friedhelm HANSEN and On the Gabriel Dimension and Sub-
Mark L. TEPLY : idealizer Rings ....................... 95
Melvin HOCHSTER: Big and Small Cohen-Macaulay Modules ........ 119
Roger WIEGAND: Rings of Bounded Module Type .................. 143
Carl FAITH: Injective Quotient Rings of Commutative Rings .... 151
Zoltan PAPP: Spectrum, Topologies and Sheaves for Left
Noetherian Rings ................................ 204
Problems
John BEACHY: Fully left bounded left Noetherian rings ........ 217
Carl FAITH: Bounded prime rings, pseudo-Frobenius rings, the
Jacobson radical of a ring ....................... 218
Melvin HOCHSTER: Commutative Noetherian local rings .......... 224
Saad MOHAMED: Continuous and dual-continuous modules ......... 226
Zoltan PAPP: Left stable left Noetherian rings ............... 228
×
Martha SMITH: Finitely generated algebras over a field ....... 229
J. Toby STAFFORD: Simple Noetherian rings .................... 230
Mark TEPLY: Subidealizers .................................... 232
Robert B. WARFIELD~ Jr.: Equivalence of matrices~ prime rings~
number of generators~ stable spaces . 235
PA PE RA
CANCELLATION FOR NONPROJECTIVE MODULES
J. T. Stafford
Brandeis University
Waltham, Massachusetts 02154
In 7 Serre showed that, given a commutative Noetherian ring R and
a projective, finitely generated R-module M with
f - rk(M) ~ dim(max(R)) + ,i then M ~ M' ~R. When Bass considered this
result in I, he was able to remove the projectivity condition. He was
further able to show that~ if M ~ R ~ N • R, then M ~ N. However for
this cancellation theorem he still had to require that M had a "large"
projective direct summand. Thus the obvious question remains as to whether
this second result still holds without the projectivity condition. In this
paper we answer this question affirmatively. This comes as a corollary of
the main results of this paper~ where we prove that the above two theorems
hold for modules over fully bounded J-Noetherian rings.
Of course, to do this we need a definition of rank that requires no
localization. For this purpose we use the r - rk of 8 (which for a
finitely generated module over a commutative Noetherian ring is equivalent
to f - rk). In fact the proofs given here closely follow those given in
8, where versions of the above two theorems were proved for noncommuta-
tive Noetherian rings. However the results given there used~ for the di-
mension on the ring, the Krull dimension of Rentschler and Gabriel 6,
which, of course, is in general larger than dim(max(R)). However it was
claimed that the methods used in 8 would answer the question posed above~
and this paper can be considered as a substantiation of that assertion.
Throughout this paper all rings will contain an identity and all
modules will be unitary.
~i. Notation and Preliminary Results.
The results of this paper hold for a more general class of rings than
Noetherian rings and we start by defining that class. The notation comes
from 3. Let R be a ring and I an ideal of R. Then I is call~d
a J-ideal (respectively a J-prime) if I is an ideal (respectively a
prime ideal) that is the intersection of the maximal ideals containing it.
The ring is called J-Noetherian if it has the ascending chain condition
on J-ideals. Define J-dim R to be the maximal length n of chains of
J-primes,
Jo ~ Jl ~ "'" ~Jn ~ R.
If R is commutative then these concepts coincide with the ones used by
Bass. In particular, R is J-Noetherian if and only if the max spectrum
of R is Noetherian, in which case J-dim R = dim(max(R)).
As mentioned in the introduction, we will prove the results of this
paper for more than just commutative rings. Define a prime ring R to
be left bounded if any essential left ideal contains a non-zero ideal.
Define a ring R to be fully left bounded Goldie if every prime factor
ring is left bounded and left Goldie. For example, commutative rings,
PI rings, and FBN rings are fully left bounded Goldie. One of the crutial
facts that we need about these rings is the following.