The further results on SEP elements in a ring with involution

In this paper, we further study many new characterizations of $SEP$ elements in a ring with involution. Firstly, combining Moore-Penrose invertible element, group invertible element, we find some $PE$ elements to characterize $SEP$ elements and then further discover some equivalent conditions for $SEP$ elements especially around the element aa^*a⁺a. Mainly, by constructing some equations in a given set including a⁺, a^*, (a^#)^*, a⁺a, aa⁺, we obtain a lot of new characterizations of $SEP$ elements. Next, we study the expression forms of related bivariate equations to depict $SEP$ elements. Finally, we use nil-cleanity of the element aa^*a⁺a to link $SEP$ elements with $PE$ elements.