Rings Close to Regular

ISBN-13: 9789048161164 / Angielski / Miękka / 2010 / 350 str.

A. a. Tuganbaev

Opis

This is the first monograph on rings closed to von Neumann regular rings. The following classes of rings are considered: exchange rings, pi-regular rings, weakly regular rings, rings with comparability, V-rings, and max rings. Every Artinian or von Neumann regular ring A is an exchange ring (this means that for every one of its elements a, there exists an idempotent e of A such that aA contains eA and (1-a)A contains (1-e)A). Exchange rings are very useful in the study of direct decompositions of modules, and have many applications to theory of Banach algebras, ring theory, and K-theory. In particular, exchange rings and rings with comparability provide a key to a number of outstanding cancellation problems for finitely generated projective modules. Every von Neumann regular ring is a weakly regular pi-regular ring (a ring A is pi-regular if for every one of its elements a, there is a positive integer n such that a is contained in aAa) and every Artinian ring is a pi-regular max ring (a ring is a max ring if every one of its nonzero modules has a maximal submodule). Thus many results on finite-dimensional algebras and regular rings are extended to essentially larger classes of rings. Starting from a basic understanding of ring theory, the theory of rings close to regular is presented and accompanied with complete proofs.

The book will appeal to readers from beginners to researchers and specialists in algebra; it concludes with an extensive bibliography.

Recenzje

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)

Spis treści

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

O autorze

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.