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

Abstract

Let $R$ be a ring and $a, d_{1},d_{2}\in R$.
First, we obtain several equivalent conditions for the equality $aa^{\|d_{1}}=a^{\|d_{2}}a$ to hold,
under the condition $a\in R^{\|d_{1}}\cap R^{\|d_{2}}$.
Then, when $a\in R^{\|\bullet d_{1}}\cap R^{\|\bullet d_{2}}$,
the equality $a^{m}a^{\|d_{1}}=a^{\|d_{2}}a^{m}$ ($m\in \mathbb{N}$) 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.