Generalized Invertibility in a Corner Ring

Abstract

Let $R$ be a ring with unity and let $a, g\in R$ be such that $a$ is regular.
In this article, the generalized invertibility of $ag+1-aa^{-}$ are investigated in term of the generalized invertibility of elements in a corner ring.
As applications, several equivalent conditions on the Drazin invertibility of product and difference of idempotents are obtained.
Moreover, we present the equivalent conditions for the existence of Moore-Penrose inverse in a ring with involution.