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 R with a unit 1 , 0. We prove that generalized inverses in R are related to idempotent group endomorphisms ρ : R → 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 R to be a principal or an annihilator ideal of an idempotent element of 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.