Let R be a semiprime ring and I a nonzero ideal of R. A map F:R→R is called a multiplicative generalized derivation if there exists a map d:R→R such that F(xy) = F(x)y+xd(y), for all x,y∈R. In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: i) $F\left(\left[u,v\right]\right)=\pm u^m\left[u,v\right]u^n,$ ii) $F\left(u\circ v\right)=\pm u^m(u\circ v)u^n,$ iii) F is SCP on I, iv) F(u)∘F(v) = u∘v, for all u,v∈I.

SOME COMMUTATIVITY THEOREMS OF PRIME RINGS WITH GENERALIZED (σ, τ)-DERIVATION prime rings;derivations;generalized derivations;generalized (${\sigma}$, ${\tau}$)-derivations;centralizing mappings; In this paper, we extend some well known results concerning generalized derivations of prime rings to a generalized (${\sigma}$, ${\tau}$)-derivation.

Let N be a semiprime right near-ring and I a semigroup ideal of N. A map f : N → N is called a multiplicative generalized (θ, θ)–derivation if there exists a multiplicative (θ, θ)–derivation d : R → R such that f(xy) = f(x)θ(y) + θ(x)d(y), for all x, y ∈ R. The purpose of this paper is to investigate the following: (i) f(uv) = ±uv, (ii) f(uv) = ±vu, (iii) f(u)f(v) = ±uv, (iv) f(u)f(v) = ±vu, (v) d(u)d(v) = θ ([u, v]), (vi) d(u)d(v) = θ (uov), (vii) d(u)θ(v) = θ(u)d(v).

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let $\mathcal{F}$ be a ring with a semiprime ideal $\Pi$. A map $\varphi :\mathcal{F}\to \mathcal{F}$ is classified as a multiplicative generalized derivation if there exists a map $\sigma :\mathcal{F}\to \mathcal{F}$ such that $\varphi \left(xy\right)=\varphi \left(x\right)y+x\sigma \left(y\right)$ for all $x,y\in \mathcal{F}$. This study focuses on semiprime ideals $\Pi$ that admit multiplicative generalized derivations $\varphi$ and G that satisfy certain differential identities within $\mathcal{F}$. 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.