Lectures on Injective Modules and Quotient Rings
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49 Carl Faith Rutgers, The State University, New Brunswick, N. J.
1967
Lectures on Injective Modules and Quotient Rings
•. f!I , "Vl
Springer-Verlag· Berlin· Heidelberg· New York
All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photornechanlcal means (photostat, microfilm and/or microcard) or by other procedure without written permission from Springer Verlag. © by Springer-Verlag Berlin' Heidelberg 1967. Library of Congress Catalog Card Number 67-31680. Printed in Germany. Title No. 7369.
- III -
TABLE OF CONTENTS
Page IV
PREFACE TO THE SPRINGER EDITION
VI
INTRODUCTION
VIII
ACKNOWLEDGEMENTS
IX
SPECIAL SYMBOLS
XI
O.
DEFINITIONS
1.
1•
INJECTIVE MODULES
13
2.
ESSENTIAL EXTENSIONS AND THE INJECTIVE HULL
22
3.
QUASI-INJECTIVE MODULES
35
4.
RADICAL AND SEMIPRIMITIVITY IN RINGS
44
5.
THE ENDOMORPHISM RING OF A QUASI-INJECTIVE MODULE
51
6.
NOETHERIAN, ARTINIAN, AND SEMI SIMPLE MODULES AND RINGS
58
7.
RATIONAL EXTENSIONS AND LATTICES OF CLOSED SUBMODULES
64
8.
MAXIMAL QUOTIENT RINGS
76
9.
SEMIPRIME RINGS WITH MAXIMUM CONDITION
82
10.
NIL AND SINGULAR IDEALS UNDER MAXIMUM CONDITIONS
86
11 •
STRUCTURE OF NOETHERIAN PRIME RINGS
96
12.
MAXIMAL QUOTIENT RINGS
105
13.
QUOTIENT RINGS AND DIRECT PRODUCTS OF FULL LINEAR RINGS
123
14.
JOHNSON RINGS
127
1 5.
OPEN PROBLEMS
132
REFERENCES
135
ADDED BIBLIOGRAPHY
138
INDEX
- IV -
PREFACE TO THE SPRINGER EDITION These Lectures were written for beginning graduate students, and for that reason are self-contained except for one or two of the chapters at the very end. The main part of the Lectures can be covered in once-weekly sessions of 75-90 minutes each running through two semesters. I have taken the opportunity to make a number of revisions, have added to the bibliography, but have not included any of the new literature on the subject. Gabriel's thesis listed in the additional bibliography has obtained the theorems of Johnson, Utumi, and Goldie, among others, functorially through the concept of localizing subcategories of abelian categories. Lambek's excellent text, Rings and Modules (Blaisdell 1966), contains some results on quotient rings not covered in these Lectures, and his comments on the literature (Ibid. pp. 166-171) are especially valuable.
SUMMARY OF REVISIONS AND ADDITIONS (1)
I have revised the treatment of the Johnson ring of quotients by first intro-
ducing Utumi's ring of quotients, using Lambek's [2] characterization, and obtaining Johnson's ring of quotients as a special case. This appears at the beginning of §8. (2)
I have revised §12 on maximal quotient rings, and also added enough material
to get the following theorem: If
R
is a prime ring, and if
quotient ring ring of
S
of
R
such that
e K
is an idempotent in the (Johnson) maximal
= eSe f"\
R
0
eSe
An example is adduced to show that when in addition
ring of
is the maximal quotient
K. R ,then
eSe
S
is the classical quotient
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