In 2012, Drazin introduced a class of outer generalized inverse in a ring R, the (b, c)-inverse of a for a, b, c ∈ R and denoted by a‖(b,c). In this paper, rank equalities of AkA‖(B,C) − A‖(B,C)Ak and (A∗)kA‖(B,C) − A‖(B,C)(A∗)k are obtained. As applications, we investigate equivalent conditions for the equalities (A∗)kA‖(B,C) = A‖(B,C)(A∗)k and AkA‖(B,C) = A‖(B,C)Ak. As corollaries we obtain rank equalities related to the Moore-Penrose inverse, the core inverse, and the Drazin inverse. The paper finishes with some rank equalities involving different expressions containing A‖(B,C),