Generalized Hirano inverses in rings

Abstract

We introduce and study a new class of generalized inverses in rings. An element a in a ring R has generalized Hirano inverse if there exists such that We prove that the generalized Hirano inverse of an element is its generalized Drazin inverse. An element has generalized Hirano inverse if and only if there exists such that . We then completely determine when a 2 × 2 matrix over projective-free rings has generalized Hirano inverse. Cline’s formula and additive properties for generalized Hirano inverses are thereby obtained.