Rings with Morita Duality
Associative rings that possess Morita dualities or self- dualities form the object of this book. They are assumed to have an identity and modules are assumed unitary. The book sets out to give an extensive introduction to thisclass of rings, covering arti
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen
1523
WeiminXue
Rings with Morita Duality
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Autor Weimin XUE Department of Mathematics Fujian Normal University Fuzhou, Fujian 350007 People's Republic of China
Mathematics Subject Classification (1991): 16-02, 16D90, 16D50, 16P20, 16P40, 16S20, 16U80 ISBN 3-540-55770-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55770-9 Springer- Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready by author/editor 46/3140-543210 - Printed on acid-free paper
This book is dedicated to Kent R. Fuller
TABLE OF CONTENTS Preface Chapter 1
1
Introduction to Morita Duality
1. Preliminaries
1
.
2. Basic Characterizations of Morita Duality
15
3. Linearly Compact Modules . . .
22
. . .
.
4. Morita Duality and Linear Compactness
31
5. Cogenerator Rings
42
.
. . . . .
. . . .
47
6. Duality of Linearly Compact Commutative Rings Chapte 2
54
Morita Duality and Ring Extensions .
54
8. Finite Triangular Extensions
58
7. Some Basic Facts
9. Finite Normalizing Extensions
65
10. Trivial Extensions
75
Chapter 3
. . . . .
Artinian Rings with Morita Duality (I)
11. Azumaya-Morita Theorem, Fuller Theorem and QF-Rings 12. Artinian Left Duo Rings Chapter 4
.
Artinian Rings II: Azumaya's Exact Rings
84 84 104 111 111
13. Azumaya·s Exact Rings 14. Exact Bimodules and Rings
117
15. Locally Distributive Rings
125
16. Artinian duo Rings . .
142
Chapter 5
. . .
Other Types of Rings with Morita Duality
17. Noetherian Rings 18. Perfect Rings and Quasi-Perfect Rings
.
. 149 149 152
Bibliography .
160
Subject Index
165
PREFACE As a generalization of the duality of vector spaces over division rings, Azumaya [59) and Morita [58) established the theory of Morita duality.
Such a duality is an additive
contravariant category equivalence between two categories of R-left- and
S-right-modules, which are both closed under Bub-
and factor modules and contain all finitely generated modules. Azumaya [59) and Morita [58) have shown that these dualities are precisely those equivalent to the functors bimodules
RES
Hom(-, E)
induced by
that are injective cogenerators on both sides and
=
=
satisfy S End(RE) and R End(E and that the natural S)' domain and ran
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