Derivations Vanishing Identities Involving Generalized Derivations and Multilinear Polynomial in Prime Rings

Abstract

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

• (i) there exist $a, b, p\in U$, $\lambda \in C$ such that $F(x)=\lambda x+bx+xa^2$, $G(x)=ax$, $d(x)=[p, x]$ for all $x\in R$ with $[p, b]=0$ and $f(x_1,\ldots , x_n)^2$ is central valued on R;

• (ii) there exist $a, b, p\in U$ such that $F(x)=ax$, $G(x)=xb$, $d(x)=[p,x]$ for all $x\in R$ and $f(x_1,\ldots , x_n)^2$ is central valued on R with $[p, a-b^2]=0$;

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

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

• (v) there exist $a, p\in U$, $\lambda , \alpha , \mu \in C$ such that $F(x)=\lambda x-a^2x$, $G(x)=xa$ and $d(x)=[p,x]$ for all $x\in R$ with $a^2=\mu -\alpha p$ and $\alpha p^2+(\lambda -2\mu ) p\in C$;

• (vi) there exist $a\in U$, $\lambda \in C$ such that R satisfies $s_4$ and either $F(x)=\lambda x+xa^2$, $G(x)=ax$ or $F(x)=\lambda x-a^2x$, $G(x)=xa$ for all $x\in R$.