Generalized inverses and their relations with clean decompositions

Abstract

An element a in a ring R is called clean if it is the sum of an idempotent e and a unit u. Such a clean decomposition a = e + u is said to be strongly clean if eu = ue and special clean if aR eR = (0). In this paper, we prove that a is Drazin invertible if and only if there exists an idempotent e and a unit u such that an = e + u is both a strongly clean decomposition and a special clean decomposition, for some positive integer n. Also, the existence of the Moore-Penrose and group inverses is related to the existence of certain - clean decompositions.