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arXiv · 2304.06149

Generalized inverses, ideals, and projectors in rings

Patricia Mariela Morillas
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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 $1 \neq 0$. We prove that generalized inverses in $\mathcal{R}$ are related to idempotent group endomorphisms $\rho: \mathcal{R} \rightarrow \mathcal{R}$, 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.

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Patricia Mariela Morillas. 2023-04-12. Generalized inverses, ideals, and projectors in rings. https://doi.org/10.2298/fil2419715m

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