New Characterizations of SEP Elements in a Ring with Involution

The characterization of SEP elements is a classical problem in generalized inverse theory. Most existing characterizations are formulated in terms of specific algebraic identities. This paper proposes a new approach based on polynomial equations with parameters to characterize SEP elements. This framework provides an alternative characterization and, more importantly, naturally clarifies the structural relationships between SEP elements and three other types of elements: square rootable elements, invertible elements, and involutional projection elements.