Generalized inverses, ideals, and projectors in rings

TL;DR

This paper explores the relationship between generalized inverses and projections in rings, providing new characterizations, existence conditions, and studying specific types of inverses in rings with involution.

Contribution

It extends the theory of generalized inverses and projectors from matrices to arbitrary rings, establishing new relations and conditions for various inverse types and ideals.

Findings

Generalized inverses relate to idempotent endomorphisms in rings.

Conditions for ideals to be principal or annihilator ideals of idempotents.

Characterizations of specific generalized inverses like Drazin and Moore-Penrose 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 $1 \neq 0$. 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.