Regular elements determined by generalized inverses

Abstract

In a semiprime ring, von Neumann regular elements are determined by their inner inverses. In particular, for elements [Formula: see text] of a von Neumann regular ring [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] denotes the set of inner inverses of [Formula: see text]. We also prove that, in a semiprime ring, the same is true for reflexive inverses.