On Hirano inverses in rings

TL;DR

This paper introduces a new class of Drazin inverses called Hirano inverses in rings, providing characterizations and exploring their properties within ring theory.

Contribution

It defines Hirano inverses, shows they coincide with Drazin inverses, and offers new characterizations in ring theory.

Findings

Hirano inverse is equivalent to Drazin inverse for elements in rings.

Provides several characterizations of Hirano inverses.

Establishes properties and conditions for the existence of Hirano inverses.

Abstract

We introduce and study a new class of Drazin inverses. An element $a$ in a ring $R$ has Drazin inverse $b$ if $a^2-ab\in N(R)$, $ab=ba$ and $b=bab$. Every Hirano inverse of an element is its Drazin inverse.We drive several characterization for this generalized inverse.