Table Of ContentDie Grundlehren der
mathematischen Wissenschaften
in Einzeldarstellungen
mit besonderer Berucksichtigung
der Anwendungsgebiete
Band 190
Herausgegeben von
J.
L. Doob . A. Grothendieck . E. Heinz· F. Hirzebruch
E. Hopf . W. Maak . S. MacLane . W. Magnus. J. K. Moser
M. M. Postnikov· F. K. Schmidt· D. S. Scott· K. Stein
GeschdftsJiihrende Herattsgeber
B. Eckmann und B. L. van der Waerden
Carl Faith
Algebra: Rings, Modules
and Categories I
Springer-Verlag Berlin Heidelberg New York 1973
Carl Faith
Rutgers University
New Brunswick, N. J., U.S.A.
Geschiiftsfiihrende Herausgeber:
B.Eckmann
EldgenBssische Technlsche Hochschule Zilrich
B. L. van der Waerden
Mathematisches Institut der Universitat Ziirich
AMS Subject Classifications (1970):
12-01,13-01,15-01,16-01,18-01,20-01
ISBN-13: 978-3-642-80636-0 e-ISBN-13 :978-3-642-80634-6
DOl: 10.10071978-3-642-80634-6
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@ by Springer·Verlag, Berlin· Heidelberg 1973.
Softcover reprint of the hardcover 1st edition 1973
Library of Congress Catalog Card Number 72-96724.
For my three muses:
Mickey, Heidi, and Cindy
Preface
This book is designed to introduce students to the basic ideas and
operations of rings, modules, and categories as patiently and as thoroughly
as time and space permit, and then bring them to the frontiers of research
as rapidly and as comprehendingly as their abilities permit. Therefore,
the reader is forewarned not to be deceived by the elementary beginning,
including a foreword on set theory and a first chapter on rudimentary
group theory and category theory.
This book is a survey of aspects of ring theory since Jacobson's Colloq
uium volume published in 1956, and revised in 1964, and all of Volume
II is devoted to ring theory. Many of the chapters are based on papers
in areas where there is intense research activity - the authors of these
papers are mentioned briefly at the end of this preface, and the papers are
discussed in the introduction - so students will encounter plenty of mo
tivation in the current literature should they wish to continue the study!
Although a rapid intellectual growth is desireable, it is intellectual
growth (period) which excites the teacher, and so ideas ought not be
given short shrift in any serious study of them. For this reason, many
important theorems are presented in the text in more than one aspect:
I have included three independent proofs of the Goldie and Lesieur
Croisot theorems on rings having semisimple right quotient rings, two
independent proofs of the Johnson-Utumi maximal right quotient rings,
three proofs of one of the most important Morita theorems, viewed
tensor products both as adjoint functors and as traditionally constructed
(ditto for the Gabriel localization theory and localization in commu
tative rings); and in many other instances for paedagogical reasons I have
offered the reader a defa vu, and, thus often a preview! This accounts
both for the dual nature of the text-interwoven as much of it is with
parts purely ring theoretical and parts purely categorical, and for the
size-two volumes in six parts. [All the category theory appears in
Volume I, however, and, as I have said, Volume II is almost wholly
(not holy!) ring theory.]
The first systematic study of the category mod-A of right modules
over a ring A was done, I believe, by Morita in 1958. In his University
VI Preface
of Oregon lectures in 1962, Bass gave simplified proofs of a number of
"Morita Theorems", incorporating ideas of Chase and Schanuel. One
of the Morita theorems characterizes when there is an equivalence of
categories mod-A mod-B for two rings A and B. Morita's solution
R::!
organizes ideas so efficiently that the classical Wedderburn-Artin theorem
is a simple consequence, and moreover, a similarity class [AJ in the
Brauer group Br(k) of Azumaya algebras over a commutative ring k
consists of all algebras B such that the corresponding categories mod-A
and mod-B consisting of k-linear morphisms are equivalent by a k-linear
functor. (For fields, Br(k) consists of similarity classes of simple central
algebras, and for arbitrary commutative k, this is subsumed under the
Azumaya [51]1 and Auslander-Goldman [60J Brauer group.) Numerous
other instances of a wedding of ring theory and category (albeit a shot
gun wedding!) are contained in the text. Furthermore, in.my attempt to
further simplify proofs, notably to eliminate the need for tensor products
in Bass's exposition, I uncovered a vein of ideas and new theorems
lying wholely within ring theory. This constitutes much of Chapter 4
-the Morita theorem is Theorem 4.29-and the basis for it is a corre
spondence theorem for projective modules (Theorem 4.7) suggested by
the Morita context. As a by-product, this provides foundation for a
rather complete theory of simple Noetherian rings-but more about
this in the introduction.
Category theory has long passed the confines of a "theory" and
truly is one of the fundamental mathematical tools, unifying many
disciplines much the way group theory does. The universality of the
concepts of morphism, functor, and natural transformation has changed
the intellectual basis of mathematical knowledge, irrevocably I believe,
from the 19th century's dependence on point-sets and functions to the
20th century. (It is rather surprising to find that categories and functors
were not introduced until the 20th was middle aged2.)
Nevertheless, I have eschewed a functorial foundation for set theory,
mainly, I suppose, because I do not think about rings and modules
functorially. Rather than solve problems in-ring theory, category theory
raises interesting questions which deepen our understanding of the
subjecP. Thus, diverse theorems of Wedderburn, Artin, Jacobson,
Johnson, Goldie, Lesieur, Croisot, Utumi, Morita, and Azumaya could
1 Azumaya [51J refers to a paper in the Bibliography published by Azumaya
in 1951. If more than one paper or book appears in one year, then small letters,
such as Jacobson [45aJ, [45bJ, and [45cJ, will distinguish them.
2 In 1945, by Eilenberg and MacLane; d. MacLane [63, p. 34J.
3 "The notions of category and functor provide not profound theorems but a
convenient language." Ibid., p. 33. (A theorem in a recent study by Mitchell [72]
may date this quotation!)
Preface VII
be subsumed under the Grothendieck-Gabriel theory of quotient abelian
categories and localizing functors in Gabriel's thesis. (More about this
in the introduction.)
It is a deceptive practice to list authors whose works are being
presented in a text, not only because some authors inevitably will be
slighted (for perhaps no other reason than that the importance of their
work has not yet been perceived by the author), but also because the
anthologized may be represented by less than his best. Nevertheless,
we have anthologized the following immortals: E. Artin, G. Burnside,
R. Dedekind, H. Fitting, G. Frobenius, C. F. Gauss, D. Hilbert, L. Kron
ecker, J.Levitzki, E. Maschke, T.Nakayama, E.Noether, O.Ore, LSchur,
J. von Neumann, and J. H. M. Wedderburn.
The major contributors to ring theory and module theory represented
in the text are: S. A. Amitsur, K. Asano, M. Auslander, G. Azumaya,
R. Baer, H. Bass, J. E. Bjork, N. Bourbaki, R. Brauer, H. Cartan,
S. U. Chase, C. Chevalley, J. H. Cozzens, R. Croisot, B. Eckmann,
A. W. Goldie, O. Goldman, M. Ikeda, N. Jacobson, R. E. Johnson,
L Kaplansky, G. Kothe, W. Krull, J. Lambek, L. Lesieur, E. Matlis,
K. Morita, H. PriHer, A. Schopf, Y. Utumi, C. L. Walker, and E. A.
Walker.
The major contributors to the category theory are: D. Buchsbaum,
S. Eilenberg, P. Freyd, P. Gabriel, A. Grothendieck, D. Kan, S. Lubkin,
S. MacLane, B. Mitchell, N. Popesco, C. Watts, and N. Yoneda.
We hasten to remark that these divisions are hardly exact! We have
commented on the categorical aspects of Morita's work, but one could
also trace the influence on category theory of ring theory, for example
the structure of rings without chain conditions of Jacobson, or the in
vention of the injective hull of Baer and Eckmann-Schopf. Similarly, the
contributions to ring theory of Eilenberg and MacLane are inestimable.
The second and third groupings are contemporary and/or recent
authors, but here again there are wide variations. Perhaps it is senseless
to make distinctions which omit two of the most influential mathe
maticians in the subject, namely P. M. Cohn and LN. Herstein. But
I can take this opportunity to note the excellent exposition of the work
of Herstein, and of his students, in the Carus Monograph [68]; and,
moreover, the work of P. M. Cohn on free ideal rings, and coproducts
of rings, is included in his book Cohn [71].
Princeton, February 1973 Carl Faith
Acknowledgements
No book ever is written by the efforts of one man alone. I wish to
thank my wife for this observation, and the amount of help she gaye
me which prompted it. I have to thank my teen-age daughters Heidi
and Cindy for bearing my great despair, forgiving my inattentions, and
denying me pity. All of which served me to get on with it.
I t is a pleasure to acknowledge other dependencies as well. Research
which eventually became a part of the text was supported by a number
of fellowships. In 1959-1960, I held a NATO Postdoctoral Fellowship
at Heidelberg, and an NSF Postdoctoral Fellowship at the Institute for
Advanced Study in 1960-1961, followed by a grant-in-aid in 1961-1962.
The main writing of the text was aided by two Rutgers Faculty Fellow
ships, one taken to the University of California, Berkeley, in 1965 -1966,
and one taken in Princeton, 1969-1970. I have to thank the Rutgers
Research Council for these.
I am deeply aware of the privileges such generous support for my
work bestows, and the concomittent obligations it exacts. The writing
of this book, the love and labor I have lavished on it, is my whole
hearted attempt to discharge these obligations, and to show my appre
ciation. I owe the faculties and staffs of the Institute for Advanced
Study, and the Mathematical Institute of Heidelberg, too many favors
to acknowledge individually, but Deane Montgomery and Friedrich Kasch
have my eternal gratitude for their help.
In addition to my former students Ann K. Boyle, Jolm H. Cozzens, and
Barbara L. Osofsky, I am obliged to many people who read parts of the
manuscript in its various editions. For many, many suggestions, which
I often incorporated in the text without specific acknowledgement, I wish
to thank Paul M. Cohn, Kent R. Fuller, Nathan Jacobson, Arun Ja
tegaonker, Serge Lang, William R. Scott, and Wolmer Vasconcelos. I
want to thank Hyman Bass for his generous permission to use portions
of his Tata Lecture Notes [67]. The material in Chapter 12 on the Morita
theorems and the Picard group, and the study of separable algebras
and the Brauer group, Chapter 32, are with minor changes, his work.
I am indebted to Richard Brauer for sending me a requested copy of
a chapter of his on Frobenius algebras, and for an inspiring communi
cation from Balestrand. Saunders MacLane raised a point on my de
finition of categories. I hope that I have resolved it correctly. For other
Acknowledgements IX
help on categories, I have leaned on my colleagues Myles Tierney and
Barry Mitchell, especially the excellent text of Mitchell [64]. My friend
Joe Rosenstein pointed out and corrected as many logical errors in the
Foreword as I would allow! I also want to thank Bill Scott for some
heart-felt advice, reaped from his own experiences in book-writing (which
I took).
I am grateful for the full-hearted cooperation of my students,
colleagues, and mathematical friends, at Rutgers and elsewhere, but,
of course, they are not to be blamed for errors! John H. Cozzens and
John Koehl can not be given too much credit for diligence in picking out
countless typos. They have my great thanks.
Mathematics typing requires a special kind of skill, not to mention
intelligence, or bravery. For courage under fire, and beyond the call of
duty, I wish to praise Ms. "Taffy" Griffith, Ms. "Euthie" Anthony, and
Ms. Caroline Underwood. Ms. Underwood has helped legions of visitors
at the Institute for Advanced Study, but what has helped me most in
my work there is her great sunniness and kindness.
The problem of assembling the bibliography accurately from hundreds
of index cards was promptly solved by my friend and neighbor, Frank
Crawford, President of Princeton Microfilm Company. He has my
finest Cordon RMtge '66.
Mathematicians have a theory about everyt hing-I have a theory
about mathematicians: so intent are we on our scribbles and our equa
tions, we are likely to be adjudged too detached, and therefore too cold
and unemotional. My theory is that it is because mathematicians are so
emotional that they can become mathematicians. They, at least, do
have the capacity to be moved by the austere beauty mathematics
possesses-the "white goddess" the poet Robert Graves spoke of as the
muse of poetry-the muse of us all.
Introduction
Grothendieck's paper [57J placed homological algebra in the setting
of abelian categories with exact direct limits (Chapter 14). Although
most of the text requires only a modicum of preparation from category
theory, and almost none from abelian categories, the far-reaching
Popesco-Gabriel theorem [62J depends not only of the foundations of
abelian categories (Chapters 5 and 6), but also very heavily on Gabriel's
thesis [62J. We give an account of this in Chapters 15 and 16, preceded
by preliminaries of Freyd's adjoint functor theorem (Chapter 5), abelian
categories (Chapter 6), and Grothendieck categories, including the Freyd
and Mitchell embedding theorems (Chapter 14). The adjointness of the
functors HomR (M, ) and ®RM, for any R-module (Kan [58J) is taken
up in Chapter 5, and again in Chapter 11, where the basic properties
of tensor products are developed in the traditional mode, and flat mo
dules introduced.
These 5 chapters on categories, nevertheless, represen,t only about a
sixth of the text, and despite the fact that the Morita theorems, and the
theorems on quotient rings, are special cases of Gabriel's localizing
functors, we prove these theorems ring-theoretically. For example,
Morita's theorem characterizing the similarity of two rings A and B,
that is, when mod-A ~ mod-B, is Theorem 29 of Chapter 4. This
happens iff there exists a finitely generated projective module P which
generates the category mod-B, and for which there is a ring isomorphism
A ~ End PB = HomB(P,P). This condition is right-left symmetric,
and we write A B. (The commonest example is B Bn , the
I"o.J I"o.J
n X n matrix ring over B.)This is preceded in Chapter 4 by a correspond
ence theorem for projective modules generalizing the Morita correspond
ence to the situation where U is a finitely generated projective faithful
left module over a ring B, and A = EndBU. Then, there is a lattice
isomorphism between the set of (right) A-submodules of U, and the right
ideals of B of the form I = IT, where T is the trace ideal of U in B.
The inverse mapping sends I = IT onto I U, and (B,A)-submodules
of U onto ideals of the form I = IT. Moreover, the correspondence
S ~ Sf of the set of ideals of A and the set of ideals of B of the form
Sf = T SfT is a lattice and semigroup isomorphism such that Sf U = US.
The main applications characterize when A is semiprime, prime, simple,
right Artinian or Noetherian, has uniform or lateral right ideals.
