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 ? R such that bab = b,b ? comm2(a),(a2-ab)k ? J(R) for some k ? N. We prove that a ? R has p-Hirano inverse if and only if there exists p = p2 ? comm2(a) such that (a2-p)k ? J(R) for some k ? N. Multiplicative and additive properties for such generalized inverses are thereby obtained. We then completely determine when a 2 x 2 matrix over local rings has p-Hirano inverse.
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