Generalized invertibility in two semigroups of a ring

• Author(s): Patrício, Pedro ; Puystjens, Roland
• Date: 2004
• Persistent ID: http://hdl.handle.net/1822/2888
• Origin: RepositóriUM - Universidade do Minho
• Subject(s): Generalized invertibility; Corner rings; Matrices over rings; Semigroups
• Document Type: Article
• Language: English

Description

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.