On ps-Drazin inverses in a ring

Abstract

An element a in a ring R has a ps-Drazin inverse if there exists $bin comm^2(a)$ such that $b=bab,{(a-ab)}^kin Jleft.Rright|$ 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−a2)k ∈ J(R) for some k ∈ N. We showCline’s formula and Jacobson’s lemma for ps-Drazin inverses. The additive properties of ps-Drazin inverses in a Banachalgebra are obtained. Moreover, we completely determine when a 2 × 2 matrix A ∈ M2(R) over a local ring R has aps-Drazin inverse.