New characterizations for $w$-core inverses in rings with involution

Abstract

Let $R$ be a unital $*$-ring and let $a,b,w\in R$. In this paper, we give some new characterizations on $w$-core inverses in $R$.  In particular, it is shown that $a$ is $w$-core invertible if and only if it is $w(aw)^{n-1}$-core invertible for any positive integer $n$, in which case, the representations of the $w$-core inverse and the $w(aw)^{n-1}$-core inverse of $a$ are both presented. We further characterize $w$-core inverses by Hermitian elements (or projections) and units.