Prime Ideals with P-Semiderivation in Rings

Abstract

In this paper, we introduce the notion of a P-semiderivation associated with a prime ideal of an arbitrary ring. We establish several basic properties of P-semiderivations and investigate commutativity conditions for the factor ring R/P. Under suitable differential identities involving commutators and Jordan products, we prove that the factor ring R/P is a commutative integral domain or the image of the P-semiderivation is contained in the prime ideal P. Our results generalize several known commutativity theorems concerning derivations and P-derivations to the broader framework of P-semiderivations.