On the Structure of Near-Rings via P-Semiderivations

This paper introduces the concept of a P-semiderivation on a near-ring N, which serves as a generalization of the classical notion of a semiderivation. We investigate the fundamental properties of P-semiderivations in relation to a nonempty subset P⊆N. The study primarily focuses on the structure of 3-prime near-rings that admit such mappings. Key results establish the conditions under which a P-semiderivation induces a classical semiderivation on the quotient near-ring N/P. Furthermore, we explore the commutativity of the quotient near-ring N/P, demonstrating that if the image of a nontrivial P-semiderivation is contained within the P-center of N, then N/P is either a commutative ring or has characteristic 2. Several examples are provided to illustrate the distinction between P-semiderivations and standard semiderivations.