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

Abstract

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.