Semiderivations on rings with certain algebraic identities

This paper presents a new type of semiderivation in ring, called a (f, g)-semiderivation, where f and g are are self-mappings of a ring. We explore various properties of (f, g)-semiderivations and demonstrate that if a semiprime ring R satisfies specific differential identities involving a (f, g)-semiderivation D, then D must be a zero map on R. Additionally, we examine the commutativity of prime rings that admit nonzero (f, g)-semiderivations which satisfy certain differential identities.