Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings

Abstract

Let $R$ be an associative prime ring of $\mathrm{char}R\ne 2$ with center $Z(R)$ and extended centroid $C$, $f(x_1,...,x_n)$ a nonzero multilinear polynomial over $C$ in $n$ noncommuting variables, $d$ a nonzero derivation of $R$ and $\rho$ a nonzero right ideal of $R$. We prove that: (i) if $[d^2(f(x_1,...,x_n)),f(x_1,...,x_n)]=0$ for all $x_1,...,x_n\in \rho$, then $\rho C=eRC$ for some idempotent element $e$ in the socle of $RC$ and $f(x_1,...,x_n)$ is central-valued in $eRCe$ unless $d$ is an inner derivation induced by $b\in Q$ such that $b^2=0$ and $b\rho =0$; (ii) if $[d^2(f(x_1,...,x_n)),f(x_1,...,x_n)]\in Z(R)$ for all $x_1,...,x_n\in \rho$, then $\rho C=eRC$ for some idempotent element $e$ in the socle of $RC$ and either $f(x_1,...,x_n)$ is central in $eRCe$ or $eRCe$ satisfies the standard identity $S_4(x_1,x_2,x_3,x_4)$ unless $d$ is an inner derivation induced by $b\in Q$ such that $b^2=0$ and $b\rho =0$.