A Note on $*$-Clean Rings

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.