Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms

• Date: 2022
• Main Authors: Tiwari, S. K., Prajapati, B.
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2022
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/6108
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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 $R.$