Rings close to regular. (English) Zbl 1120.16012

Tuganbaev, Askar

Mathematics and its Applications (Dordrecht) 545. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0851-1/hbk). xii, 350 p. (2002).

This is the first monograph on rings close 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 each of its elements \(a\), there is a positive integer \(n\) such that \(a^n\) 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. (Publisher’s description)
The book will appeal to readers from beginners to researchers and specialists in algebra; it concludes with an extensive bibliography.
The seven chapters of the book are the following: 1. Some basic facts of ring theory, 2. Regular and strongly regular rings, 3. Rings of bounded index and \(I_0\) rings, 4. Semiregular and weakly regular rings, 5. Max rings and \(\pi\)-regular rings, 6. Exchange rings and modules, 7. Separative exchange rings.

Reviewer: Xue Weimin (Fujian)

MSC:

16E50	von Neumann regular rings and generalizations (associative algebraic aspects)
16D70	Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16-02	Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16E20	Grothendieck groups, \(K\)-theory, etc.

Keywords:

von Neumann regular rings; exchange rings; \(\pi\)-regular rings; weakly regular rings; rings with comparability; \(V\)-rings; max-rings; idempotents; direct decompositions of modules