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

TL;DR

This paper provides a concise proof of Cline's formula for Drazin invertibility and strong cleanness in rings, along with new characterizations of certain elements in corner rings, enhancing understanding of invertibility properties.

Contribution

It offers a new, shorter proof of Cline's formula and extends results to strongly clean elements and corner ring characterizations.

Findings

If $ab$ is Drazin invertible, then $ba$ is Drazin invertible.

Strongly clean property of $ab$ implies the same for $ba$.

Characterizations of Drazin invertibility in corner rings.

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.