Algebra II Ring Theory
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Editors
S.S.Chern J.L.Doob J.Douglas,jr. A. Grothendieck E. Heinz F. Hirzebruch E. Hopf S. Mac Lane W. Magnus M. M. Postnikov F.K.Schmidt W.Schmidt D.S.Scott K. Stein J. Tits B. L. van der Waerden Managing Editors B. Eckmann
J. K. Moser
Carl Faith
Algebra II Ring Theory
Springer-Verlag Berlin Heidelberg New York 1976
Carl Faith Rutgers, The State University, New Brunswick, N.J. 08903 and The Institute for Advanced Study, Princeton, N.J. 08540, USA
AMS Subject Classifications (1970): 12-01, 13-01, 15-01, 16-01, 18-01, 20-01 ISBN-13: 978-3-642-65323-0 DOl: 10.1007/978-3-642-65321-6
e-ISBN-13: 978-3-642-65321-6
Library of Congress Cataloging in Publication Data. Faith, Carl Clifton, 1927. ~ Algebra. Bibliography: v. 1, p. Contents: 1. Rings, modules and categories. - 2. Ring theory. 1. Rings (Algebra) 2. Modules (Algebra) 3. Categories (Mathematics). I. Title. II. Series: Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 191. QA247.F 34. 512'.4. 72-96724. 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 1976 Softcover reprint of the hardcover 1st edition 1976
This volume is for my family
Mickey, Heidi, and Cindy Eldridge, Louise, and Frederick Virginia Nell Caudill Compton Harold Compton 1895-1964 and the memory of my parents
Herbert Spencer Faith 1895-1952 Vila Belle Foster Faith 1897-1965
Preface to Volume II
I. Ring Theory The term The Theory of Rings seems first used as title of a book by Jacobson [43], and in his preface Jacobson asserts that the theory that forms the subject of the book had its beginning with Artin's extension in 1927 of Wedderburn's structure theory of algebras to rings satisfying the chain conditions. 1 As the predecessor to his book, Jacobson cites Deuring's Algebren (Deuring [35, 68]), and Deuring cites Dickson's Algebren und ihre Zahlentheorie (ZUrich, 1927). As in his earlier book, Algebra and its Arithmetic, Dickson ([23]) extends arithmetic in algebraic number field, that is, arithmetic of the ring of integers in a finite field extension k of the field X
= Y or
x= Z,
'tfy,ZEL.
11
17. Modules of Finite Length and their Endomorphism Rings
The next theorem was proved by E. Noether [21] for ideals of a commutative nng. 17.4 Proposition (Kurosch [35], Ore [36]). If L is a modular lattice, then any two irredundant finite intersections of an element aEL as an intersection (meet) of irreducible elements, say,
have the same number of elements r=s. Moreover, for any subset I of {I, ... , n}, there is an injection p: 1- {I, ... , n} such that
(replacement lemma) is an irredundant int
