Let R be a ring and a, d 1 , d 2 ∈ R. First, we obtain several equivalent conditions for the equality aa ∥d 1 = a ∥d 2 a to hold, under the condition a ∈ R ∥d 1 ∩ R ∥d 2. Then, when a ∈ R ∥ • d 1 ∩ R ∥ • d 2 , the equality a m a ∥d 1 = a ∥d 2 a m (m ∈ N) is also investigated by means of Drazin inverses. Next, some characterizations for the invertibility of aa ∥d 1 − a ∥d 2 a are obtained. Particularly, a number of examples are given to illustrate our results.