Further results on the EP-ness and co-EP-ness involving Mary inverses

Let R be a ring and a, d1,d2 is an element of R. First, we obtain several equivalent conditions for the equality aa parallel to d1 = a parallel to d2a to hold, under the condition a is an element of R parallel to d1 boolean AND R parallel to d2. Then, when a is an element of R parallel to center dot d1 boolean AND R parallel to center dot d2, the equality ama parallel to d1 = a parallel to d2am (m is an element of N) is also investigated by means of Drazin inverses. Next, some characterizations for the invertibility of aa parallel to d1 - a parallel to d2a are obtained. Particularly, a number of examples are given to illustrate our results.