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Rings of Quotients

An Introduction to Methods of Ring Theory

  • Book
  • © 1975

Overview

Authors:
  1. Bo Stenström

    1. Matematiska Institutionen, Stockholms Universitet, Sweden

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Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 217)

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Table of contents (17 chapters)

  1. Front Matter

    Pages I-VIII
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  2. Introduction

    • Bo Stenström
    Pages 1-3

  3. Notations and Conventions

    • Bo Stenström
    Pages 4-4

  4. Modules

    • Bo Stenström
    Pages 5-49

  5. Rings of Fractions

    • Bo Stenström
    Pages 50-62

  6. Modular Lattices

    • Bo Stenström
    Pages 63-81

  7. Abelian Categories

    • Bo Stenström
    Pages 82-113

  8. Grothendieck Categories

    • Bo Stenström
    Pages 114-135

  9. Torsion Theory

    • Bo Stenström
    Pages 136-159

  10. Hereditary Torsion Theories for Noetherian Rings

    • Bo Stenström
    Pages 160-178

  11. Simple Torsion Theories

    • Bo Stenström
    Pages 179-194

  12. Rings and Modules of Quotients

    • Bo Stenström
    Pages 195-212

  13. The Category of Modules of Quotients

    • Bo Stenström
    Pages 213-224

  14. Perfect Localizations

    • Bo Stenström
    Pages 225-243

  15. The Maximal Ring of Quotients of a Non-Singular Ring

    • Bo Stenström
    Pages 244-261

  16. Finiteness Conditions on Mod-(A, ℑ)

    • Bo Stenström
    Pages 262-272

  17. Self-Injective Rings

    • Bo Stenström
    Pages 273-282

  18. Classical Rings of Quotients

    • Bo Stenström
    Pages 283-294

  19. Back Matter

    Pages 295-312
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Keywords

About this book

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 papers appeared (by R. E. Johnson, Y. Utumi, A. W. Goldie, P. Gabriel, J. Lambek, and others). Since then the progress has been rapid, and the subject has by now attained a stage of maturity, where it is possible to make a systematic account of it (which is the purpose of this book). The most immediate example of a ring of quotients is the field of fractions Q of a commutative integral domain A. It may be characterized by the two properties: (i) For every qEQ there exists a non-zero SEA such that qSEA. (ii) Q is the maximal over-ring of A satisfying condition (i). The well-known construction of Q can be immediately extended to the case when A is an arbitrary commutative ring and S is a multiplicatively closed set of non-zero-divisors of A. In that case one defines the ring of fractions Q = A [S-l] as consisting of pairs (a, s) with aEA and SES, with the declaration that (a, s)=(b, t) if there exists UES such that uta = usb. The resulting ring Q satisfies (i), with the extra requirement that SES, and (ii).

Authors and Affiliations

  • Matematiska Institutionen, Stockholms Universitet, Sweden

    Bo Stenström

Bibliographic Information

  • Book Title: Rings of Quotients

  • Book Subtitle: An Introduction to Methods of Ring Theory

  • Authors: Bo Stenström

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-3-642-66066-5

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1975

  • Softcover ISBN: 978-3-642-66068-9Published: 22 December 2011

  • eBook ISBN: 978-3-642-66066-5Published: 06 December 2012

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: 309

  • Topics: Algebra

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