Acta Mathematica Vietnamica

Let $R$ be a prime ring of $\mathrm{char(R)}\not=2$, $U$ its Utumi ring of quotients and center $C=Z(U)$ its extended centroid, $I$ a both sided ideal of $R$, $f(x_1,\dots,x_n)$ a multilinear polynomial over $C$, that is noncentral-valued on $R, F, G$ be two generalized derivations of $R$ and $d$ be a derivation of $R$. Let $f(I)$ be the set of all evaluations of the multilinear polynomial $f(x_1,\dots,x_n)$ in $I$. If $$d(F^2(u)u)=uG^2(u)$$ for all $u\in f(I)$, then all possible forms of the maps are determined. As an application of this result, we also study the commutator identity $[F^2(u)u, G^2(v)v]=0$ for all $u, v\in f(I)$, where $F$ and $G$ are two generalized derivations of $R$.

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