Characterizations of weighted core inverse in rings with involution

TL;DR

This paper characterizes and represents the weighted core inverse in rings with involution using idempotents and units, extending previous results and providing new conditions for invertibility and EP properties.

Contribution

It introduces new characterizations of weighted core inverse and weighted-EP elements in rings with involution, generalizing existing results with novel conditions involving idempotents and invertible Hermitian elements.

Findings

Characterization of $e$-core invertibility via idempotents and invertibility conditions.

Conditions for weighted-EP elements with respect to $(e,f)$ involving idempotents.

Generalization and improvement of previous results in ring theory.

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, $n\geqslant 1$, 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}.