Characterizations of weighted core inverse in rings with involution

Abstract

$R$ is a unital ring with involution. We investigate the characterizations and representations of weighted core inverse of an element in $R$ by idempotents and units. For example, let $a\in R$ and $e\in R$ be an invertible Hermitian element, $n⩾1$, then $a$ is $e$-core invertible if and only if there exists an element (or an idempotent) $p$ such that $(ep{)}^{\ast }=ep$, $pa=0$ and $a^n+p$ (or $a^n(1-p)+p$) is invertible. As a consequence, let $e,f\in R$ be two invertible Hermitian elements, then $a$ is weighted-$\mathrm{EP}$ with respect to $(e,f)$ if and only if there exists an element (or an idempotent) $p$ such that $(ep{)}^{\ast }=ep$, $(fp{)}^{\ast }=fp$, $pa=ap=0$ and $a^n+p$ (or $a^n(1-p)+p$) is invertible. These results generalize and improve conclusions in \cite{Li}.