Commutativity and Centrality Conditions Induced By Generalized Skew Derivations on Prime and Semiprime Near-Rings

Abstract

In this paper, we investigate the influence of generalized skew derivations on the structural properties of prime and semiprime near-rings. In particular, we establish several new commutativity and centrality conditions arising from differential identities involving generalized skew derivations associated with automorphisms. By extending classical derivation techniques to a broader near-ring framework, we obtain sufficient conditions under which prime near-rings become commutative and semiprime near-rings exhibit centralizing behavior. The study further examines the interaction between generalized skew derivations, Lie ideals, and annihilator conditions in 2-torsion free algebraic structures. A number of new theorems are proved concerning the behavior of generalized skew derivations satisfying certain algebraic identities on ideals and subsets of prime and semiprime near-rings. These results generalize and unify several well-known commutativity theorems previously established for ordinary derivations, skew derivations, and generalized derivations in rings and near-rings. Moreover, the obtained results demonstrate that generalized skew derivations impose strong algebraic restrictions on noncommutative structures, thereby providing a broader operator-theoretic framework for studying centrality and commutativity in generalized algebraic systems. The findings contribute to the ongoing development of derivation theory in noncommutative algebra and open new directions for future investigations involving Lie ideals, multiplicative derivations, Γ-near-rings, and related operator identities