Left core inverses in rings with involution

Abstract

We study the left version of core inverses in $*$-rings. An example is given to show that there exists a $*$-ring in which a left $(a, a^{*})$-invertible element a need not be regular. Based on this fact, the left core inverse which is different from the left $(a, a^{*})$-inverse is introduced and studied. For any $*$-ring R and $a, x \in R$, x is called a left core inverse of a if $axa=a, xax=x,(ax)^{*}=ax$ and $xa^2=a$. It is proved that x is a left core inverse of a if and only if x is both a left $(a,a^{*})$-inverse and a $\{1\}$-inverse of a.