Identities with Generalized Derivations and Multilinear Polynomials

Abstract

Let R be a prime ring of characteristic different from 2, U be its Utumi quotient ring with extended centroid C and $f(x_1,\ldots ,x_n)$ be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations on R such that $F^2(f(r))f(r)-G(f(r)^2)=0$ for all $r=(r_1,\ldots ,r_n)\in R^{n}$. Then one of the following holds:

• (i) there exists $a\in U$ such that $F(x)=xa$ and $G(x)=a^2x$ for all $x\in R$ with $a^2\in C$;

• (ii) there exists $a\in U$ such that $F(x)=ax$ and $G(x)=a^2x$ for all $x\in R$;

• (iii) $f(r_1,\ldots ,r_n)^2$ is central valued on R and one of the following holds:

  • (a) there exist $a, b\in U$ such that $F(x)=xa$ and $G(x)=xa^2+[b,x]$ for all $x\in R$ with $a^2\in C$;

  • (b) there exist $a,b\in U$ such that $F(x)=ax$ and $G(x)=xa^2+[b,x]$ for all $x\in R$.