A Note on $*$-Clean Rings

TL;DR

This paper explores properties of $*$-clean rings, introduces the new class strongly $oldsymbol{ ext{pi}}$-$*$-regular rings, and characterizes their structure and stable range conditions.

Contribution

It introduces the concept of strongly $oldsymbol{ ext{pi}}$-$*$-regular rings and provides characterizations and properties related to $*$-clean rings.

Findings

$*$-ring $R$ is strongly $oldsymbol{ ext{pi}}$-$*$-regular iff $R$ is $oldsymbol{ ext{pi}}$-regular with all idempotents as projections.

$R$ is strongly $oldsymbol{ ext{pi}}$-$*$-regular iff $R/J(R)$ is strongly regular with $J(R)$ nil.

Stable range conditions of $*$-clean rings are discussed and equivalent conditions are established.

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.