Exploring Prime Near-Rings and Semiprime Rings Under Skew and Generalized Skew Derivations

Abstract: This paper investigates the structural properties of prime near-rings and prime rings under the action of skew derivations and generalized skew derivations. We examine how these mappings influence algebraic identities, commutativity conditions, and the behavior of ideals within such algebraic systems. By establishing new characterizations and commutativity criteria, the study extends classical results on prime rings to a broader framework that includes near-rings. Furthermore, we demonstrate how generalized skew derivations unify and extend several well-known derivation concepts, thereby providing a more flexible approach to analyzing algebraic structures. These findings contribute to the deeper understanding of prime near-rings and semiprime rings, offering new tools for further research in ring theory and its applications.