J. Korean Math. Soc. 2015; 52(4): 839-851
Printed July 1, 2015
https://doi.org/10.4134/JKMS.2015.52.4.839
Copyright © The Korean Mathematical Society.
Jian Cui and Zhou Wang
Anhui Normal University, Southeast University
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
MSC numbers: 16W10, 16U99
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