On Hirano inverses in rings

Abstract

We completely characterize a subclass of Drazin inverses by means of tripotents and nilpotents. We provethat an element a in a ring R has Hirano inverse if and only if a2 2 R has strongly Drazin inverse, if and only if a􀀀a3is nilpotent. If 122 R, we prove that a 2 R has Hirano inverse if and only if there exists p3 = p 2 comm2(a) such thata 􀀀 p 2 N(R) , if and only if there exist two idempotents e; f 2 comm2(a) such that a + e 􀀀 f 2 N(R) . Multiplicativeand additive results for this generalized inverse are thereby obtained.