$*$ $(g,e)$ quasi-normal ring and dual $*$ $(g,e)$ quasi-normal ring

Abstract

Let $R$ be a $*$-ring and $g,e\in E(R)$, the set of idempotents of $R$. The ring $R$ is said to be a (resp. dual) $*$ $(g,e)$ quasi-normal ring if $gae=0$ implies (resp. $g^*Rae=0$) $gaRe^*=0$. We prove that $R$ is (resp. dual) $*$ $(g,e)$ quasi-normal if and only if (resp. $g^*R(1-g)Re=0$) $gR(1-e)Re^*=0$. As by-products, we give a $*$-ring, which is clean, almost clean, $*$-clean, almost $*$-clean, $*$-regular and unit regular. Moreover, we use some matrix rings to describe (dual) $*$ $(g,e)$ quasi-normal rings. Finally, we consider the relations between (dual) $*$ $(g,e)$ quasi-normal rings and other generalized inverses.