Pair of generalized derivations acting on multilinear polynomials in prime rings

Abstract

Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$ and $f(r_1,\ldots,r_n)$ be a multilinear polynomial over $C$, which is not central valued on $R$. Suppose that $F$ and $G$ are two nonzero generalized derivations of $R$ such that $G\neq Id$ (identity map) and $$F(f(r)^2)=F(f(r))G(f(r))+G(f(r))F(f(r))$$ for all $r=(r_1,\ldots,r_n)\in R^n$. Then one of the following holds:(1) there exist $\lambda \in C$ and $\mu \in C$ such that $F(x)=\lambda x$ and $G(x)=\mu x$ for all $x\in R$ with $2\mu=1$;(2) there exist $\lambda \in C$ and $p,q\in U$ such that $F(x)=\lambda x$ and $G(x)=px+xq$ for all $x\in R$ with $p+q\in C$, $2(p+q)=1$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;(3) there exist $\lambda \in C$ and $a\in U$ such that $F(x)=[a,x]$ and $G(x)=\lambda x$ for all $x\in R$ with $f(x_1,\ldots,x_n)^2$ is central valued on $R$;(4) there exist $\lambda \in C$ and $a,b\in U$ such that $F(x)=ax+xb$ and $G(x)=\lambda x$ for all $x\in R$ with $a+b\in C$, $2\lambda =1$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;(5) there exist $a, p\in U$ such that $F(x)=xa$ and $G(x)=px$ for all $x\in R$, with $(p-1)a=-ap\in C$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;(6) there exist $a, q\in U$ such that $F(x)=ax$ and $G(x)=xq$ for all $x\in R$ with $a(q-1)=-qa\in C$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$.