Commutative Rings whose Finitely Generated Modules Decompose
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723 Willy Brandal
Commutative Rings whose Finitely Generated Modules Decompose
Springer-Verlag Berlin Heidelberg New York 1979
Author
Willy Brandal Department of Mathematics University of Tennessee Knoxville, TN 37916/USA
AMS Subject Classifications (1970): 13-02, 13 C05, 13 F05, 13 F10, 13G05 ISBN 3-540-09507-1 ISBN 0-387-09507-1
Springer-Verlag Berlin Heidelberg NewYork Springer-Verlag NewYork Heidelberg Berlin
Library of Congress Cataloging in Publication Data. Brandal, Willy, 1942- The commutative rings whose finitely generated modules decompose. (Lecture notes in mathematics ; v. 723) Bibliography: p. Includes index. 1. Commutative rings. 2. Modules (Algebra) 3. Decomposition (Mathematics) I. Title. II. Series: Lecture notes in mathematics (Berlin) ; v. 723. QA3.L28 no. 723 [QA251.3] 510'.8s [512'.4] 79-15959 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 t979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Table of Contents Introduction Part I 9 14 23 29 37 44 49 58 64
Section Section Section Section Section Section Section Section Section
1 2 3 4 5 6 7 8 9
Linearly Compact Modules and ~Imost Maximal Rings h-local Domains Valuation Rings and Bezout Rings Basic Facts About FGC Rings and the Local Case Further Facts About FGC Rings and Torch Rings The Zariski and Patch Topologies of the Spectrum of a Ring The Stone-Cech Compactification of N Relating Topology to the Decomposition of Modules The Main Theorem Part I I
72 82 88 97 98 108 110 113
Section Section Section Section Section Section
Proving the Main Theorem
Constructing Examples
10 Valuations 11 Long Power Series Rings 12 Maximally Complete Valuation Domains 13 Examples of Maximal Valuation Rings 14 Examples of Almost Maximal Bezout Domains 15 Examples of Torch Rings Bibliography Index of Notation and D e f i n i t i o n s
Introduction Throughout a l l rings w i l l be commutative with i d e n t i t i e s , a l l modules w i l l be u n i t a r y modules, and R w i l l always denote a r i n g . ring i f every f i n i t e l y
R is said to be an FGC
generated R-module decomposes i n t o a d i r e c t sum of c y c l i c
submodules. The purpose of these notes is to describe a l l the FGC r i n g s ; i . e . , t e r i z e the FGC rings and give as many examples as possible.
charac-
One form of the
Fundamental Theorem of Abelian Groups says that the ring of integers is an FGC ring.
Another form of t h i s theorem says that
P.I.D.'s
are FGC rings.
Thus
we present a generalization of the Fundamental Theorem of Abelian Group
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