Co-EP-Ness and EP-Ness Involving the Inverse along an Element

Let a,d be two elements in rings and a∥d be the inverse of a along d. When a∥d exists, we obtain several characterizations for the invertibility of aa∥d−a∥da, which is related to the invertibility of elements expressed by certain functions of a,d and suitable elements from the center of the ring. On the other hand, some equivalent conditions for the equality aa∥d=a∥da, as the complement of the previous invertibility in some sense, are given by means of the group inverses and the ring units, respectively. Then, the results obtained are applied in a ∗-ring, namely, when d=a∗, the co-EP and EP properties are deduced correspondingly.