Rings Close to Regular

  • Askar Tuganbaev

Part of the Mathematics and Its Applications book series (MAIA, volume 545)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Askar Tuganbaev
    Pages 1-66
  3. Askar Tuganbaev
    Pages 67-112
  4. Askar Tuganbaev
    Pages 113-152
  5. Askar Tuganbaev
    Pages 153-186
  6. Askar Tuganbaev
    Pages 187-228
  7. Askar Tuganbaev
    Pages 229-278
  8. Askar Tuganbaev
    Pages 279-314
  9. Back Matter
    Pages 315-350

About this book

Introduction

Preface All rings are assumed to be associative and (except for nilrings and some stipulated cases) to have nonzero identity elements. A ring A is said to be regular if for every element a E A, there exists an element b E A with a = aba. Regular rings are well studied. For example, [163] and [350] are devoted to regular rings. A ring A is said to be tr-regular if for every element a E A, there is an element n b E A such that an = anba for some positive integer n. A ring A is said to be strongly tr-regular if for every a E A, there is a positive integer n with n 1 n an E a + An Aa +1. It is proved in [128] that A is a strongly tr-regular ring if and only if for every element a E A, there is a positive integer m with m 1 am E a + A. Every strongly tr-regular ring is tr-regular [38]. If F is a division ring and M is a right vector F-space with infinite basis {ei}~l' then End(MF) is a regular (and tr-regular) ring that is not strongly tr-regular. The factor ring of the ring of integers with respect to the ideal generated by the integer 4 is a strongly tr-regular ring that is not regular.

Keywords

DEX Exchange Finite K-theory Maxima algebra eXist maximum proof ring ring theory

Authors and affiliations

  • Askar Tuganbaev
    • 1
  1. 1.Moscow Power Engineering InstituteTechnological UniversityMoscowRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-015-9878-1
  • Copyright Information Springer Science+Business Media B.V. 2002
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-6116-4
  • Online ISBN 978-94-015-9878-1
  • About this book