Rings of Quotients An Introduction to Methods of Ring Theory
The theory of rings of quotients has its origin in the work of (j). Ore and K. Asano on the construction of the total ring of fractions, in the 1930's and 40's. But the subject did not really develop until the end of the 1950's, when a number of important
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S.S.Chern J.L.Doob J.Douglas,jr. A. Grothendieck E. Heinz F. Hirzebruch E. Hopf W. Maak S. Mac Lane W. Magnus M. M. Postnikov F. K. Schmidt W. Schmidt D. S. Scott K. Stein Geschiiftsfiihrende Herausgeber B. Eckmann
J. K. Moser B. L. van der Waerden
Bo Stenstrom
Rings of Quotients An Introduction to Methods of Ring Theory
Springer-Verlag Berlin Heidelberg New York 1975
Bo Stenstrom Matematiska Institutionen, Stockholms Universitet
AMS Subject Classifications (1910): 16-01, 16-02, 16A08, 18-01, 18 E 15, 18 E40
ISBN- I 3: 978-3-642-66068-9 001: 10.1007/978-3-642-66066-5
e-ISBN-13: 918-3-642-66066-5
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, [C·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 maOe for other than private use, a fee is payable to the publisher, the amount of the fee to he determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelherg 1975. Softcover reprint of the hardcover 1st edition 1975 Library of Congress Cataloging in Publication Data. Stenstrom, Bo.T. Rings of quotients. (Die Grundlebren der mathematischen Wissenschaften in Einzeldarstellungen: Bd.217.) Bibliography: p. Includes index. I. Quotient rings. 2. Torsion theory (Algebra). 3. Categories (Mathematics). I. Title. II. Series. QA251.5.S8. 512'.55. 75-1003.
Contents
1
Introduction . . . . . . . Notations and Conventions.
4
Chapter I. Modules . . . .
5
§ 1. Basic Definitions. . . . § 2. Sums and Products of Modules § 3. Finitely Generated Modules and Noetherian Modules. § 4. Categories and Functors . . . . . . . . . . . § 5. Exactness of Functors between Module Categories § 6. Projective and Injective Modules. § 7. Semi-Simple Modules and Rings. § 8. Tensor Products. § 9. Bimodules . . . . § 10. Flat Modules. . . § 11. Pure Submodules . § 12. Regular Rings . § 13. Coherent Rings. . Exercises. . . . . . .
5 8
10 13 16 19 23
26 31 33
36 39 41
44
Chapter II. Rings of Fractions.
50
§ 1. The Ring of Fractions . . § 2. Orders in a Semi-Simple Ring § 3. Modules of Fractions. . . . § 4. Invertible Ideals and Hereditary Orders . Exercises. . . . . . . . . .
50 54 57 59 61
Chapter III. Modular Lattices .
63
§ 1. Lattices . . . . . . . . § 2. Modularity . . . . . . .
63 65
§ 3. Lattices with Chain Condition. § 4. Distributive Lattices . . . . . § 5. Continuous Lattices . . . . . § 6. Pseudo-Complemented Lattices § 7. Closure Operators . § 8. Galois Connections Exercises. . . . . . .
66
69 72
74 76 77 80
VI
Chapter IV. Abelian Categories
Contents
82
§ 1. Equivalence of Categories . § 2. Kernels and Cokernels . . § 3. Products and Coproducts . § 4. Abelian Categories. . . . § 5. Pullbacks and Pushouts . § 6. Generators and Cogenerators § 7. Functor Categories. § 8. Limits and Colimits . § 9. Adjoint Functors . . § 10. Morita Equivalence
