A NOTE ON STRONGLY *-CLEAN RINGS

Title & Authors
A NOTE ON STRONGLY *-CLEAN RINGS
CUI, JIAN; WANG, ZHOU;

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 $\small{{\pi}}$-*-regular are introduced. It is shown that R is strongly $\small{{\pi}}$-*-regular if and only if R is $\small{{\pi}}$-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 $\small{{\pi}}$-*-regular ring;stable range condition;
Language
English
Cited by
1.
Some characterizations of ∗-regular rings, Communications in Algebra, 2017, 45, 2, 841
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