On ps-Drazin inverses in a ring

Abstract

An element a in a ring R has a ps-Drazin inverse if there exists b ∈ comm 2 (a) such that b = bab, (a − ab) k ∈ J(R) for some k ∈ N . Elementary properties of ps-Drazin inverses in a ring are investigated here. We prove that a ∈ R has a ps-Drazin inverse if and only if a has a generalized Drazin inverse and (a − a 2 ) k ∈ J(R) for some k ∈ N . We show Cline's formula and Jacobson's lemma for ps-Drazin inverses. The additive properties of ps-Drazin inverses in a Banach algebra are obtained. Moreover, we completely determine when a 2 × 2 matrix A ∈ M2(R) over a local ring R has a ps-Drazin inverse.