Generalized Derivations on Prime Near-Rings and Commutativity Conditions

Abstract

In this paper, we present a systematic investigation of derivations and generalized derivations on prime near-rings, focusing on their influence on the underlying algebraic structure. We examine several annihilation conditions involving derivations and generalized derivations and show that, in a prime near-ring, any element that annihilates the image of a non-zero derivation or its associated generalized derivation must be zero. This result establishes a strong rigidity phenomenon for derivations in prime near-rings. Furthermore, we analyze the interaction between generalized derivations and commutators and prove that certain vanishing or annihilation conditions involving commutators force the near-ring to be commutative. These findings demonstrate that derivations and generalized derivations function not only as algebraic operators but also as effective tools for detecting and controlling non-commutative behavior in near-rings. As a consequence, our results extend several classical commutativity theorems from ring theory to the broader framework of near-rings, thereby contributing to the structural theory of prime near-rings.