P-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 p-Hirano inverse
if there exists b \in R such that bab = b; b \in comm^2(a); (a^2 -
ab)k \in J(R) for some k \in N. We prove that a \in R has p-
Hirano inverse if and only if there exists p = p^2 \in comm^2(a)
such that (a^2-p)^k \in J(R) for some k \in N. Multiplicative and
additive properties for such generalized inverses are thereby
obtained. We then completely determine when a 2\times 2 matrix
over local rings has p-Hirano inverse.