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© 2002

Rings Close to Regular

Book

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

  1. 1.Moscow Power Engineering InstituteTechnological UniversityMoscowRussia

About the authors

Askar Tuganbaev received his Ph.D. at the Moscow State University in 1978 and has been a professor at Moscow Power Engineering Institute (Technological University) since 1978. He is the author of three other monographs on ring theory and has written numerous articles on ring theory.

Bibliographic information

  • Book Title Rings Close to Regular
  • Authors A.A. Tuganbaev
  • Series Title Mathematics and Its Applications
  • 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
  • Hardcover ISBN 978-1-4020-0851-1
  • Softcover ISBN 978-90-481-6116-4
  • eBook ISBN 978-94-015-9878-1
  • Edition Number 1
  • Number of Pages XII, 350
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Associative Rings and Algebras
  • Buy this book on publisher's site

Reviews

From the reviews:

"This is the first monograph on rings close to von Neumann regular rings. … The book will appeal to readers from beginners to researchers and specialists in algebra; it concludes with an extensive bibliography." (Xue Weimin, Zentralblatt MATH, Vol. 1120 (22), 2007)