EP elements and ∗−Strongly Regular Rings

Abstract

Let $R$ be a ring with involution $*$. An element $a\in R$ is called $*-$strongly regular if there exists a projection $p$ of $R$ such that $p\in comm^2(a)$, $ap=0$ and $a+p$ is invertible, and $R$ is said to be $*-$strongly regular if every element of $R$ is $*-$strongly regular. We discuss the relations among strongly regular rings, $*-$strongly regular rings, regular rings and $*-$regular rings. Also, we show that an element $a$ of a $*-$ring $R$ is $*-$strongly regular if and only if $a$ is $EP$. Hence we finally give some characterizations of $EP$ elements.