Properties of generalized strongly Drazin invertible elements in general rings

Abstract

In this paper, we define the generalized strong Drazin inverse in a general ring and investigate this class of inverses. Thus, recent results on the strong Drazin invertible and generalized strong Drazin invertible elements are extended to a more general setting. In particular, we show that [Formula: see text] is generalized strong Drazin invertible in a general ring [Formula: see text] if and only if there exists an idempotent [Formula: see text] such that [Formula: see text] and [Formula: see text] is quasinilpotent in [Formula: see text]. We also prove that if [Formula: see text] is generalized Drazin invertible in [Formula: see text] for some [Formula: see text], so are [Formula: see text], [Formula: see text], [Formula: see text]. This partially answer to a question posed by Mosić.