Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms

Keywords: Jordan homomorphism, generalized derivations, multilinear polynomials, extended centroid, Utumi quotient ring.

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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Published
09.08.2022
How to Cite
Tiwari, S. K., and B. Prajapati. “Generalized Derivations Acting on Multilinear Polynomials As a Jordan Homomorphisms”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 7, Aug. 2022, pp. 991 - 1003, doi:10.37863/umzh.v74i7.6108.
Section
Research articles