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

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