Characterizations and representations of $w$-core inverses in rings

Abstract

Let $R$ be an associate ring with involution and let $a,w\in R$. The notion of EI along an element is introduced. An element $w$ is called EI along $a$ if $w^{\parallel a}$ exists and $w^{\parallel a}w=ww^{\parallel a}$. Its several characterizations are given by $w$-core inverses. Several necessary and sufficient conditions such that $a_w^{\tiny{\textcircled{\#}}}aw$ and $wa_w^{\tiny{\textcircled{\#}}}a$ are projections are derived. In particular, it is shown that $a_w^{\tiny{\textcircled{\#}}}aw$ is a projection if and only if $aw$ is Moore-Penrose invertible with $(aw)^\dag=a_w^{\tiny\textcircled{\tiny{\#}}}$. Also, $wa_w^{\tiny{\textcircled{\#}}}a$ is a projection if and only if $a$ is Moore-Penrose invertible with $a^\dag=wa_w^{\tiny\textcircled{\tiny{\#}}}$. Then, we describe the existence of $w$-core inverse of $a$ by the existence of (the unique) projection $p\in R$ and idempotent $q\in R$ satisfying $pR = aR = awR=qR$ and $Rq = Raw$.