Rings Close to Regular
- Editore:
Springer Netherlands
- Collana:
- Mathematics And Its Applications
- EAN:
9781402008511
- ISBN:
1402008511
- Pagine:
- 368
- Formato:
- Hardback
- Lingua:
- Inglese
Descrizione Rings Close to Regular
This is the first monograph on rings closed to von Neumannregular rings. The following classes of rings are considered: exchangerings, φ -regular rings, weakly regular rings, rings withcomparability, V-rings, and max rings. Every Artinian or von Neumannregular ring "A" is an exchange ring (this means that for everyone 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 thestudy of direct decompositions of modules, and have many applicationsto theory of Banach algebras, ring theory, and K-theory. Inparticular, exchange rings and rings with comparability provide a keyto a number of outstanding cancellation problems for finitelygenerated projective modules. Every von Neumann regular ring is aweakly regular φ -regular ring (a ring "A" is φ -regular iffor every one of its elements "a," there is a positive integer"n" such that "a" is contained in "aAa") and everyArtinian ring is a φ -regular max ring (a ring is a max ring ifevery one of its nonzero modules has a maximal submodule). Thus manyresults on finite-dimensional algebras and regular rings are extendedto essentially larger classes of rings. Starting from a basicunderstanding of ring theory, the theory of rings close to regular ispresented and accompanied with complete proofs.The book will appeal to readers from beginners to researchers andspecialists in algebra; it concludes with an extensive bibliography.
