Characterizations of SEP elements using the projectivity in a ring with involution

In this paper, we systematically investigate the characterization of SEP elements in an involution ring, using the solvability of an equation in a specific set as a necessary and sufficient criterion. Building on this, we establish a series of characterizations by analyzing the relationships between SEP elements and the followings: the projectivity of generalized inverse products, the projectivity of generalized inverse differences, and the representation of such products as differences of two p