Communications of the
Korean Mathematical Society CKMS

Let $R$ be a noncommutative prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C$ the extended centroid of $R$ and $f(x_1,\ldots,x_n)$ a noncentral multilinear polynomial over $C$ in $n$ noncommuting variables. Denote by $f(R)$ the set of all the evaluations of $f(x_1,\ldots,x_n)$ on $R$. If $d$ is a nonzero derivation of $R$ and $G$ a nonzero generalized derivation of $R$ such that $$ d(G(u)u)\in Z(R)$$ for all $u \in f(R)$, then $f(x_1,\ldots,x_n)^2$ is central-valued on $R$ and there exists $b\in U$ such that $G(x) = bx$ for all $x\in R$ with $d(b) \in C$. As an application of this result, we investigate the commutator $[F(u)u$, $G(v)v]\in Z(R)$ for all $u, v\in f(R)$, where $F$ and $G$ are two nonzero generalized derivations of $R$.

Keywords: prime ring, derivation, generalized derivation, extended centroid, Utumi ring of quotients

MSC numbers: 16W25, 16N60