Generalizations of partial isometries, EP and normal elements in rings

Abstract

Let $a,d$ be two elements in rings and $a^{\|d}$ be the inverse of $a$ along $d$. Some characterizations of the equality $a^{\|d}=d$ are given in rings. We also investigate the equivalent conditions for $aa^{\|d}=a^{\|d}a$ to hold, as well as $ad=da$. We prove that $a^{\|d}=d$ if and only if $ad$ is idempotent, when $a\in R^{\|d}$; $aa^{\|d}=a^{\|d}a$ if and only if there exists $t\in R^{-1}$ such that $a^{\|d}=at=ta$, when $a\in R^{\|\bullet d}$; $ad=da$ if and only if $a^{\|d}(a+d)=(a+d)a^{\|d}$, when $a\in R^{\|\bullet d}$. Thus, some well-known results on partial isometries, EP and normal elements in rings are extended to more general settings.