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TitleGeneralized invertibility in two semigroups of a ring
Author(s)Patrício, Pedro
Puystjens, Roland
KeywordsGeneralized invertibility
Corner rings
Matrices over rings
Issue date15-Jan-2004
JournalLinear Algebra and its Applications
Citation"Linear Algebra and its Applications". ISSN 0024-3795. 377 (2004) 125-139.
Abstract(s)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.
Publisher version
AccessOpen access
Appears in Collections:CMAT - Artigos em revistas com arbitragem / Papers in peer review journals

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