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Serial Rings

  • Book
  • © 2001

Overview

Authors:
  1. Gennadi Puninski

    1. Department of Mathematics, Moscow State Social University, Moscow, Russia

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-ix
    Download chapter PDF

  2. Basic Notions

    • Gennadi Puninski
    Pages 1-19

  3. Finitely Presented Modules over Serial Rings

    • Gennadi Puninski
    Pages 20-33

  4. Prime Ideals in Serial Rings

    • Gennadi Puninski
    Pages 34-40

  5. Classical Localizations in Serial Rings

    • Gennadi Puninski
    Pages 41-54

  6. Serial Rings with the A.C.C. on annihilators and Nonsingular Serial Rings

    • Gennadi Puninski
    Pages 55-63

  7. Serial Prime Goldie Rings

    • Gennadi Puninski
    Pages 64-79

  8. Noetherian Serial Rings

    • Gennadi Puninski
    Pages 80-92

  9. Artinian Serial Rings

    • Gennadi Puninski
    Pages 93-112

  10. Serial Rings with Krull Dimension

    • Gennadi Puninski
    Pages 113-122

  11. Model Theory for Modules

    • Gennadi Puninski
    Pages 123-135

  12. Indecomposable Pure Injective Modules over Serial Rings

    • Gennadi Puninski
    Pages 136-150

  13. Super-Decomposable Pure Injective Modules over Commutative Valuation Rings

    • Gennadi Puninski
    Pages 151-162

  14. Pure Injective Modules over Commutative Valuation Domains

    • Gennadi Puninski
    Pages 163-171

  15. Pure Projective Modules over Nearly Simple Uniserial Domains

    • Gennadi Puninski
    Pages 172-186

  16. Pure Projective Modules over Exceptional Uniserial Rings

    • Gennadi Puninski
    Pages 187-197

  17. Σ-Pure Injective Modules over Serial Rings

    • Gennadi Puninski
    Pages 198-205

  18. Endomorphism Rings of Artinian Modules

    • Gennadi Puninski
    Pages 206-210

  19. Back Matter

    Pages 211-226
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Keywords

About this book

The main theme in classical ring theory is the structure theory of rings of a particular kind. For example, no one text book in ring theory could miss the Wedderburn-Artin theorem, which says that a ring R is semisimple Artinian iffR is isomorphic to a finite direct sum of full matrix rings over skew fields. This is an example of a finiteness condition which, at least historically, has dominated in ring theory. Ifwe would like to consider a requirement of a lattice-theoretical type, other than being Artinian or Noetherian, the most natural is uni-seriality. Here a module M is called uni-serial if its lattice of submodules is a chain, and a ring R is uni-serial if both RR and RR are uni-serial modules. The class of uni-serial rings includes commutative valuation rings and closed under homomorphic images. But it is not closed under direct sums nor with respect to Morita equivalence: a matrix ring over a uni-serial ring is not uni-serial. There is a class of rings which is very close to uni-serial but closed under the constructions just mentioned: serial rings. A ring R is called serial if RR and RR is a direct sum (necessarily finite) of uni-serial modules. Amongst others this class includes triangular matrix rings over a skew field. Also if F is a finite field of characteristic p and G is a finite group with a cyclic normal p-Sylow subgroup, then the group ring FG is serial.

Authors and Affiliations

  • Department of Mathematics, Moscow State Social University, Moscow, Russia

    Gennadi Puninski

Bibliographic Information

  • Book Title: Serial Rings

  • Authors: Gennadi Puninski

  • DOI: https://doi.org/10.1007/978-94-010-0652-1

  • Publisher: Springer Dordrecht

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media Dordrecht 2001

  • Hardcover ISBN: 978-0-7923-7187-8Published: 31 August 2001

  • Softcover ISBN: 978-94-010-3862-1Published: 10 October 2012

  • eBook ISBN: 978-94-010-0652-1Published: 06 December 2012

  • Edition Number: 1

  • Number of Pages: IX, 226

  • Topics: Associative Rings and Algebras, Algebra, Mathematical Logic and Foundations, Mathematics, general

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