Generalized Hirano inverses in rings

TL;DR

This paper introduces the generalized Hirano inverse in rings, expanding the concept of inverses with specific algebraic properties and conditions involving quasinilpotent elements.

Contribution

It defines and explores properties of the generalized Hirano inverse, a new class of inverse in ring theory, with conditions involving commutativity and quasinilpotency.

Findings

Characterization of elements with generalized Hirano inverse

Conditions under which the inverse exists in rings

Connections to existing inverse concepts

Abstract

We introduce and study a new class of generalized inverse in rings. An element $a$ in a ring $R$ has generalized Hirano inverse if there exists some $b\in R$ such that $bab=b, b\in comm^2(a), a^2-ab \in R^{qnil}$