# Rings Close to Regular

Book

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

1. Front Matter
Pages i-xii
Pages 1-66
Pages 67-112
Pages 113-152
Pages 153-186
Pages 187-228
Pages 229-278
Pages 279-314
9. Back Matter
Pages 315-350

### 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

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
• Publisher Name Springer, Dordrecht
• eBook Packages
• 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
• 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)