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http://dx.doi.org/10.4134/JKMS.2015.52.4.839

A NOTE ON STRONGLY *-CLEAN RINGS  

CUI, JIAN (Department of Mathematics Anhui Normal University)
WANG, ZHOU (Department of Mathematics Southeast University)
Publication Information
Journal of the Korean Mathematical Society / v.52, no.4, 2015 , pp. 839-851 More about this Journal
Abstract
A *-ring R is called (strongly) *-clean if every element of R is the sum of a projection and a unit (which commute with each other). In this note, some properties of *-clean rings are considered. In particular, a new class of *-clean rings which called strongly ${\pi}$-*-regular are introduced. It is shown that R is strongly ${\pi}$-*-regular if and only if R is ${\pi}$-regular and every idempotent of R is a projection if and only if R/J(R) is strongly regular with J(R) nil, and every idempotent of R/J(R) is lifted to a central projection of R. In addition, the stable range conditions of *-clean rings are discussed, and equivalent conditions among *-rings related to *-cleanness are obtained.
Keywords
(strongly) *-clean ring; (strongly) clean ring; strongly ${\pi}$-*-regular ring; stable range condition;
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