Abstract: Firstly, the Cline’s formula and Jacobson’s lemma for generalized n-strong Drazin inverse were given under the conditions of acd=dbd, bdb=bac. Secondly, the equivalent characterizations of generalized n-strong Drazin inverse idempotent elements equality were given. Finally, the equivalent characterizations of generalized n-strong Drazin invertible element similarity and the problem of multi-element similarity under the condition of acd=dbd, dba=aca were discussed.

Key words: generalized n-strong Drazin inverse, similar, unit regular ring