$*$ $q$-central idempotent and $*$ quasi-normal ring

Abstract

We generalize the concept of $q$-central idempotent introduced by Lam in \cite{Lam}: an idempotent $e$ in a $*$-ring $R$ is called $*$ quarter-central (or $*$ $q$-central shortly) if $e^*R(1-e)Re^*=0$. It is proved that $e\in E(R)$ is $*$ $q$-central if and only if $e^*$ is $*$ $q$-central if and only if $eR(1-e)^*Re=0$, where $E(R)$ stands for the set of all idempotents in $R$. We show that any $*$ $q$-central idempotent must be $q$-central. However, we find examples to state that the converse of the conclusion is not true. Naturally, we give some equivalent conditions to claim when a $q$-central idempotent is $*$ $q$-central. A $*$-ring is said to $*$ quasi-normal if $e^*R(1-e)Re^*=0$ for any $e\in E(R)$. Furthermore, we list some new characterizations of $*$ quasi-normal ring.