A class of Drazin inverses in rings

A strongly direct sum decomposition given by a strongly nil-clean endomorphism motivates us to introduce a class of new Drazin inverses, which is called strong Drazin inverse. In this paper, some basic properties of the strong Drazin inverse are obtained. We show the Cline’s Formula and Jacobson’s Lemma for the strong Drazin inverse. Further, some additive results for the strong Drazin inverse are presented under additional conditions that ab = 0 or ab = ba.