Rings and Modules of Quotients
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237 80 Stenstrom University of Stockholm, Stockholm/Sweden
Rings and Modules of Quotients
,· t ,
.
Springer-Verlag Berlin· Heidelberg· NewYork 1971
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich
237 80 Stenstrom University of Stockholm, Stockholm/Sweden
Rings and Modules of Quotients
,· t ,
.
Springer-Verlag Berlin· Heidelberg· NewYork 1971
AMS Subject Classifications (1970): 16A 08
ISBN 3-540-05690-4 Springer-Verlag Berlin . Heidelberg . New York ISBN 0-387-05690-4 Springer-Verlag New York· Heidelberg· Berlin 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 1971. Library of Congress Catalog Card Number 70-180692. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach
Contents
Chapter 1. Torsion theory 1.
Preradioals
1
2.
Torsion theories
4
3.
Topologies
12
4. 5. 6.
Stable torsion theories
20
Topologies for a oommutative noetherian ring
23
!-injective modules
29
Chapter 2. Categories of modules of quotients 7.
Construction of rings and modules of quotients
8.
Modules of quotients and !-injective envelopes
41
9.
Coreflective suboategories of
44
10.
Giraud subcategories and the Popescu-Gabriel
Mod-A
theorem
33
48
Chapter 3. General properties of rings of quotients 11.
Lattioes of !-pure submodules
58
12.
Finiteness oonditions on topologies
68
13.
Flat epimorphisms of rings
12
14.
Maximal flat epimorphic extension of a ring
82
15.
I-topologies and rings of fractions
86
Chapter 4. Se1f-injeotive rings 16.
The endomorphism ring of an injeotive module
93
17.
Coperfect rings
91
18.
Quasi-Frobenius rings
101
Chapter 5. Maximal and classical rings of quotients 19.
The maximal ring of quotients
20.
The maximal ring of quotients of a non-singular ring
110 113
21.
The maximal ring of quotients of a reduced ring
118
22.
The olassioal ring of quotients
123
References
130
Introduction These notes are intended to give a survey 01 the basic, more or less well-known, results in the theory of rings of quotients.
An effort has been made to make the account as self-contained and elementary as possible. Thus we assume from the reader only a knowledge of the elements of the theory of rings and of abelian categories. We will briefly describe the contents of the notes. Chapter 1 treats the necessary preliminaries on torsion theory. The main result here is the establishing of a 1-1 correspondence between hereditary torsion theories and topologies on a ring (Gabriel [31] and Maranda
(51]).
Tn Chapter 2 we constr
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