# Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings

### Basudeb Dhara

Belda College, Paschim Medinipur, India### Rajendra K. Sharma

Indian Institute of Technology, New Delhi, India

## Abstract

Let $R$ be an associative prime ring of $charR=2$ 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 $ρ$ 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 $x_{1},...,x_{n}∈ρ$, then $ρC=eRC$ 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 $b∈Q$ such that $b_{2}=0$ and $bρ=0$; (ii) if $[d_{2}(f(x_{1},...,x_{n})),f(x_{1},...,x_{n})]∈Z(R)$ for all $x_{1},...,x_{n}∈ρ$, then $ρC=eRC$ 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 $b∈Q$ such that $b_{2}=0$ and $bρ=0$.

## Cite this article

Basudeb Dhara, Rajendra K. Sharma, Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings. Rend. Sem. Mat. Univ. Padova 121 (2009), pp. 243–257

DOI 10.4171/RSMUP/121-15