Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms
Abstract
UDC 512.5
Let $R$ be a prime ring whose characteristic is not equal to $2,$ let $U$ be the Utumi quotient ring of $R,$ and let $C$ be the extended centroid of $R.$ Also let $G$ and $H$ be two generalized derivations on $R$ and let $f(x_1,\ldots,x_n)$ be a noncentral multilinear polynomial over $C.$ If $G(H(u^2))=(H(u))^2$ for all $u=f(r_1,\ldots,r_n),$ $r_1,\ldots,r_n \in R,$ then one of the following holds:
1) $H=0;$
2) there exists $\lambda\in C$ such that $G(x)=H(x)=\lambda x$ for all $x\in R;$
3) there exist $\lambda\in C$ and $a\in U$ such that $H(x)=\lambda x$ and $G(x)=[a, x]+\lambda x$ for all $x\in R$ and $f(x_1,\ldots,x_n)^2$ is central-valued on $R.$
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