On Hirano inverses in rings

Abstract

We completely characterize a subclass of Drazin inverses by means of tripotents and nilpotents. We prove that an element $a$ in a ring $R$ has Hirano inverse if and only if $a^2\in R$ has strongly Drazin inverse, if and only if $a-a^3$ is nilpotent. If $\frac{1}{2}\in R$, we prove that $a\in R$ has Hirano inverse if and only if there exists $p^3=p\in comm^2(a)$ such that $a-p\in N(R)$, if and only if there exist two idempotents $e,f\in comm^2(a)$ such that $a+e-f\in N(R)$. Multiplicative and additive results for this generalized inverse are thereby obtained.