Generalized inverses and their relations with clean decompositions

Abstract

An element [Formula: see text] in a ring [Formula: see text] is called clean if it is the sum of an idempotent [Formula: see text] and a unit [Formula: see text]. Such a clean decomposition [Formula: see text] is said to be strongly clean if [Formula: see text] and special clean if [Formula: see text]. In this paper, we prove that [Formula: see text] is Drazin invertible if and only if there exists an idempotent [Formula: see text] and a unit [Formula: see text] such that [Formula: see text] is both a strongly clean decomposition and a special clean decomposition, for some positive integer [Formula: see text]. Also, the existence of the Moore–Penrose and group inverses is related to the existence of certain ∗-clean decompositions.