The relative properties of (b,c)-inverses are studied with respect to a given element d in a ring R. The new concepts of left and right (bd,cd)-inverses are introduced in a more general setting. It's shown that the class of right (bd,cd)-inverses is a new subclass of (b,c)-inverses, and an example is given to show that (b,c)-invertible elements need not be right (bd,cd)-invertible. Various properties including Jacobson's lemma, Cline's formula and the intertwining property for right (bd,cd)-invertible elements are studied. As applications, a new characterization of abelian rings from the point of view of right (bd,cd)-inverses is given.