Regular elements determined by generalized inverses

TL;DR

This paper investigates how regular elements in semiprime rings can be uniquely identified by their sets of inner or reflexive inverses, providing new characterizations within ring theory.

Contribution

It establishes that in semiprime rings, regular elements are uniquely determined by their inner and reflexive inverses, extending known results in ring theory.

Findings

Regular elements are characterized by their inner inverses.

Reflexive inverses also uniquely determine elements in semiprime rings.

The results apply specifically to von Neumann regular and semiprime rings.

Abstract

In a semiprime ring, von Neumann regular elements are determined by their inner inverses. In particular, for elements $a,b$ of a von Neumann regular ring $R$, $a=b$ if and only if $I(a)=I(b)$, where $I(x)$ denotes the set of inner inverses of $x\in R$. We also prove that, in a semiprime ring, the same is true for reflexive inverses.