Algebra Rings, Modules and Categories I
VI 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 R::! mod-B for two rings A an
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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 This work is subject to copyright. All rights are reserved, whether the whole or part of the material Is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. @ 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 Colloquium 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 motivation 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 LesieurCroisot 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
