Commutator identity involving generalized derivations on multilinear polynomials

Abstract

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U, C be the extended centroid of $R,\, F$ and G be two nonzero generalized derivations of R and $f(x_1,\ldots ,x_n)$ be a multilinear polynomial over C which is not central valued on R. If

$$\begin{aligned} {[}F(u)u, G(v)v]=0 \end{aligned}$$

for all $u,v\in f(R)$, then there exist $a,b\in U$ such that $F(x)=ax$ and $G(x)=bx$ for all $x\in R$ with $[a, b]=0$ and $f(x_1,\ldots ,x_n)^2$ is central valued on R.