Let R be a ring and a, d1, d2 ? R. First, we obtain several equivalent
conditions for the equality aa?d1 = a?d2 a to hold, under the condition a ?
R?d1 ? R?d2 . Then, when a ? R??d1 ? R??d2 , the equality ama?d1 = a?d2 am
(m ? N) is also investigated by means of Drazin inverses. Next, some
characterizations for the invertibility of aa?d1 ? a?d2 a are obtained.
Particularly, a number of examples are given to illustrate our results.