Some Identities Related to Semiprime Ideal of Rings with Multiplicative Generalized Derivations

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let F be a ring with a semiprime ideal Π. A map ϕ:F→F is classified as a multiplicative generalized derivation if there exists a map σ:F→F such that ϕ(xy)=ϕ(x)y+xσ(y) for all x,y∈F. This study focuses on semiprime ideals Π that admit multiplicative generalized derivations ϕ and G that satisfy certain differential identities within F. By examining these conditions, the paper aims to provide new insights into the structural aspects of rings, particularly their commutativity in relation to the behavior of such derivations.