The $w$-core inverse of a product and its applications

Abstract

Let $R$ be a unital $*$-ring. This paper establishes necessary and sufficient conditions for the $w$-core inverse of a regular element by units in $R$. Then, the existence criterion of the $w$-core inverse of the product of three elements are derived. As applications, the existence criterion and expression of the $w$-core inverse of $2\times 2$ matrices over a ring are given.