Identities with generalized derivations on multilinear polynomials in prime rings

Abstract

Let R be a prime ring of characteristic not equal to 2, U the Utumi quotient ring of R, $C=Z(U)$ the extended centroid of R, and $f(x_1,\ldots ,x_n)$ be a multilinear polynomial over C, not central valued on R. Suppose that F and G are two nonzero generalized derivations of R such that $F(f(x_1,\ldots ,x_n))G(f(x_1,\ldots ,x_n))+G(f(x_1,\ldots ,x_n))F(f(x_1,\ldots ,x_n))=0$ for all $x_1,\ldots ,x_n\in R$. Then $f(x_1,\ldots ,x_n)^2$ is central valued on R and one of the following holds:

• (1) there exist $0\ne \lambda \in C$ and $p\in U-C$, such that $F(x)=\lambda x$ for all $x\in R$ and $G(x)=[p,x]$ for all $x\in R$;

• (2) there exist nonzero $a, b \in U$, such that $F(x)=xa$ for all $x\in R$ and $G(x)=bx$ for all $x\in R$ with $ab=-ba\in C$;

• (3) there exist nonzero $a, b \in U$, such that $F(x)=ax$ for all $x\in R$ and $G(x)=xb$ for all $x \in R$ with $ab=-ba\in C$;

• (4) there exist $0\ne \lambda \in C$ and $a\in U-C$, such that $F(x)=[a,x]$ for all $x\in R$ and $G(x)=\lambda x$ for all $x\in R$.