Characterizations of <i>SEP</i> 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 projections. Furthermore, based on the generalized invertibility of projective elements, we derive two new necessary and sufficient conditions for identifying SEP elements.