Generalized invertibility in two semigroups of a ring

Abstract

In {\em Linear and Multilinear Algebra}, 1997, Vol.43, pp.137-150, R. Puystjens and R. E. Hartwig proved that given a regular element $t$ of a ring $R$ with unity $1$, then $t$ has a group inverse if and only if

$u=t^{2}t^{-}+1-tt^{-}$ is invertible in $R$ if and only if

$v=t^{-}t^{2}+1-t^{-}t$ is invertible in $R$. There, R. E. Hartwig posed the pertinent question whether the inverse of $u$ and $v$ could be directly related. Similar equivalences appear in the characterization of Moore-Penrose and Drazin invertibility, and therefore analogous questions

arise. We present a unifying result to answer these questions not only

involving classical invertibility, but also some generalized inverses as

well.