Continuous and Discrete Modules

✍ Scribed by Saad H. Mohamed, Bruno J. Müller

• Publisher: Cambridge University Press
• Year: 1990
• Tongue: English
• Leaves: 137
• Series: London Mathematical Society - Lecture Notes 147
• Category: Library

✦ Synopsis

Continuous and discrete modules are, essentially, generalizations of infective and projective modules respectively. Continuous modules provide an appropriate setting for decomposition theory of von Neumann algebras and have important applications to C*-algebras. Discrete modules constitute a dual concept and are related to number theory and algebraic geometry: they possess perfect decomposition properties. The advantage of both types of module is that the Krull-Schmidt theorem can be applied, in part, to them. The authors present here a complete account of the subject and at the same time give a unified picture of the theory. The treatment is essentially self-contained, with background facts being summarized in the first chapter. This book will be useful therefore either to individuals beginning research, or the more experienced worker in algebra and representation theory.

✦ Table of Contents

Cover......Page 1
Publications List......Page 2
Title......Page 4
ISBN 978-0-521-39975-3......Page 5
PREFACE......Page 6
TABLE OF CONTENTS......Page 10
1. A-INJECTIVE MODULES......Page 12
2. QUASI-INJECTIVE MODULES......Page 18
3. EXCHANGE AND CANCELLATION PROPERTIES......Page 19
4. DECOMPOSITION THEOREMS......Page 23
COMMENTS......Page 27
1. BASIC PROPERTIES......Page 29
2. DIRECT SUMS OF QUASI-CONTINUOUS MODULES......Page 32
3. DECOMPOSITIONS OF QUASI-CONTINUOUS MODULES......Page 35
4. INTERNAL CANCELLATION PROPERTY......Page 44
5. QUASI-CONTINUITY VERSUS QUASI-INJECTIVITY......Page 46
COMMENTS......Page 48
1. ENDOMORPHISM RINGS......Page 50
2. CONTINUOUS MODULES......Page 57
3. THE EXCHANGE PROPERTY......Page 59
COMMENTS......Page 64
1. DEFINITIONS AND BASIC RESULTS......Page 66
2. DECOMPOSITION THEOREMS......Page 70
3. APPLICATIONS OF THE DECOMPOSITION THEOREMS......Page 73
4. DISCRETENESS AND PROJECTIVITY......Page 78
5. QUASI-DISCRETENESS OF DIRECT SUMS......Page 85
COMMENTS......Page 90
1. DISCRETE MODULES......Page 92
2. ENDOMORPHISM RINGS......Page 93
3. COMMUTATIVE NOETHERIAN RINGS......Page 97
COMMENTS......Page 105
1. VARJANTS OF SUPPLEMENTATION......Page 106
2. SUPPLEMENTS ARE SUMMANDS......Page 109
3. EXTENDING MODULES......Page 110
4. THE HISTORICAL ORIGIN OF THE CONCEPT OF CONTINUITY......Page 111
5. Ro -CONTINUOUS RINGS AND MODULES......Page 114
6. OPEN PROBLEMS......Page 116
BIBLIOGRAPHY......Page 119
NOTATION......Page 133
INDEX......Page 135