Rings Close to Regular / Edition 1

by A.A. Tuganbaev

Overview

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.

Product Details

ISBN-13:	9781402008511
Publisher:	Springer Netherlands
Publication date:	09/30/2002
Series:	Mathematics and Its Applications , #545
Edition description:	2002
Pages:	350
Product dimensions:	6.14(w) x 9.21(h) x 0.36(d)

About the Author

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.

Table of Contents

1 Some Basic Facts of Ring Theory.- 2 Regular and Strongly Regular Rings.- 3 Rings of Bounded Index and I0-rings.- 4 Semiregular and Weakly Regular Rings.- 5 Max Rings and—-regular Rings.- 6 Exchange Rings and Modules.- 7 Separative Exchange Rings.

Editorial 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)

From the Publisher