Serial Rings

  • Gennadi Puninski

Table of contents

  1. Front Matter
    Pages i-ix
  2. Gennadi Puninski
    Pages 1-19
  3. Gennadi Puninski
    Pages 20-33
  4. Gennadi Puninski
    Pages 34-40
  5. Gennadi Puninski
    Pages 41-54
  6. Gennadi Puninski
    Pages 64-79
  7. Gennadi Puninski
    Pages 80-92
  8. Gennadi Puninski
    Pages 93-112
  9. Gennadi Puninski
    Pages 113-122
  10. Gennadi Puninski
    Pages 123-135
  11. Gennadi Puninski
    Pages 198-205
  12. Gennadi Puninski
    Pages 206-210
  13. Back Matter
    Pages 211-226

About this book

Introduction

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.

Keywords

Finite Morphism algebra commutative property endomorphism ring model theory ring theory

Authors and affiliations

  • Gennadi Puninski
    • 1
  1. 1.Department of MathematicsMoscow State Social UniversityMoscowRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-010-0652-1
  • Copyright Information Kluwer Academic Publishers 2001
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-94-010-3862-1
  • Online ISBN 978-94-010-0652-1
  • About this book