A note on clean elements and inverses along an element

Abstract

Let $R$ be an associative ring with unity 1 and let $a,d \in R$. An element $a \in R$ is called
invertible along $d$ if there exists unique $a^{\parallel d}$ such that $a^{\parallel d}ad=d=daa^{\parallel d}$ and $a^{\parallel d}\in dR \cap Rd$ (see \cite[Definition 4]{Mary2011}). In this note, we present new characterizations for the existence of $a^{\parallel d}$ by a clean decomposition of $ad$ and $da$. As applications, existence criteria for the Drazin inverse and the group inverse are given. Also, we show that $ad$ and $da$ are uniquely strongly clean, provided that $a^{\parallel d}$ exists and $ad=da$.