A note on unit inner inverses

In a prime ring [Formula: see text], it is proved that the nonzero unit-regular elements are determined by their unit inner inverses. More precisely, for any nonzero unit-regular elements [Formula: see text] and [Formula: see text] in a prime ring [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] denotes the set of all unit inner inverses of a unit-regular element [Formula: see text]. Several consequences and a generalization of this result are discussed.