Generalized inverses, ideals, and projectors in rings

Abstract

The theory of generalized inverses of matrices and operators is closely connected with projections, i.e., idempotent (bounded) linear transformations. We show that a similar situation occurs in any associative ring $\mathcal{R}$ with a unit $1\ne 0$. We prove that generalized inverses in $\mathcal{R}$ are related to idempotent group endomorphisms $\rho :\mathcal{R}\to \mathcal{R}$, called projectors. We use these relations to give characterizations and existence conditions for $\{1\}$, $\{2\}$, and $\{1,2\}$-inverses with any given principal/annihilator ideals. As a consequence, we obtain sufficient conditions for any right/left ideal of $\mathcal{R}$ to be a principal or an annihilator ideal of an idempotent element of $\mathcal{R}$. We also study some particular generalized inverses: Drazin and $(b,c)$ inverses, and $(e,f)$ Moore-Penrose, $e$-core, $f$-dual core, $w$-core, dual $v$-core, right $w$-core, left dual $v$-core, and $(p,q)$ inverses in rings with involution.

Version 4: Version published in the open access journal FILOMAT. It has minor corrections. In the previous version, Corollary 4.9 appears as a consequence of Theorems 4.5-4.8. In this new version, this corollary does not appear because it is an immediate consequence of the modified Corollary 2.16(2)

References in corpus (1)

• Expressions and characterizations for the Moore-Penrose inverse