Torsion Theories, Additive Semantics, and Rings of Quotients
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Series: Forschungsinstitut fUr Mathematik, ETH ZUrich
177 Joachim Lambek McGill University, Montreal, Canada
Torsion Theories, Additive Semantics, and Rings of Quotients With an appendix by H. H. Storrer on Torsion Theories and Dominant Dimensions
Springer-Verlag Berlin· Heidelberg· NewYork 1971
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and 8. Eckmann, ZUrich
Series: Forschungsinstitut fUr Mathematik, ETH ZUrich
177 Joachim Lambek McGill University, Montreal, Canada
Torsion Theories, Additive Semantics, and Rings of Quotients With an appendix by H. H. Storrer on Torsion Theories and Dominant Dimensions
Springer-Verlag Berlin· Heidelberg· NewYork 1971
ISBN 3-540-05340-9 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05340-9 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 70-148538 Printed in Germany.
Offserdruck: Julius Beltz, Weinheim/Bergstr.
CONTENTS
O.
An exposition of torsion theories ...•.......•..•.•.•.••....
1.
Additive semantics
15
2.
Torsion ideals and rings of quotients •......••••....•...•.•
32
3.
Protorsion modules ••....•...••....•.•..••.••••.•...•••.••••
54
4.
Embedding theorems.........................................
64
References.....................................................
79
Appendix: Torsion Theories and Dominant Dimension •......•...••.
83
v In Section
°
we give an expository account of tor sion theories
and equivalent concepts.
In 1958, Findlay and the author had stu-
died the binary relation between modules Band C which asserts that Hom (B, I(C}) =0, where I(C) is the injective hull of C.
A "tor
sion theory" consists of two classes of modules, the torsion modules and the torsionfree modules, satisfying the following conditions: B is torsion
Hom(B, liC)) =0 for all torsionfree C,
Cis torsionfree
¢>
Hom(B, I(C}) =0 for all torsion modules B.
Every module M has a largest torsion submodule T(Jvil. du Ia M is called "divisible" if I(Ivi)
1M is torsionfree.
has a divisible hull D(M) defined by D(j\tl)
A mo
Every module
1M = Trl(t\1) 1M).
To every
module there is assigned a torsionfree divisible module Q(M}=D(M/T(M)) which best approximates it, its "quotient module". While Section 1 is essentially independent of Section 0, it was inspired by this question: why is the quotient module Q(RR} not just a module but a ring? The answer to this question led to a theory of additive semantics.
Roughly speaking,
this tells
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