On Cline's Formula for some certain elements in a ring

Abstract

In \cite{C, LCC}, it is proven that if an element $ab$ in a ring is (generalized) Drazin invertible, then so is $ba$. In this paper, we give a new and short proof of it in an effective manner. In particular, we show that if $ab$ is strongly clean, then so is $ba$. Consequently, we see that if $1-ab$ is strongly clean, then so is $1-ba$. Also, some characterizations are obtained for some certain elements in a corner ring. It is shown that for an idempotent $e$ and any arbitrary element $a$ in a ring, $ea+1-e$ is Drazin invertible if and only if $eae$ is Drazin invertible.