TURKISH JOURNAL OF MATHEMATICS, vol.43, no.5, pp.2114-2124, 2019 (SCI-Expanded)
An element a in a ring R has a ps-Drazin inverse if there exists b is an element of comm(2) (a) such that b = bab, (a - ab)(k) J(R) for some k is an element of N. Elementary properties of ps-Drazin inverses in a ring are investigated here. We prove that a is an element of R has a ps-Drazin inverse if and only if a has a generalized Drazin inverse and (a - a(2))(k) is an element of J(R) for some k is an element of 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 x 2 matrix A is an element of M-2 (R) over a local ring R has a ps-Drazin inverse.