### 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.

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