Properties of generalized strongly Drazin invertible elements in general rings

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 a is generalized strong Drazin invertible in a general ring I if and only if there exists an idempotent p is an element of I such that p is an element of comm(2)(a) and a - p is quasinilpotent in I. We also prove that if ( ab)(k) is generalized Drazin invertible in I for some k is an element of N, so are ab, ba, (ba)(k). This partially answer to a question posed by Mosic.