Generalized Derivations and Multilinear Polynomials in Prime Rings

Basudeb Dhara ; Shuliang Huang ; Atanu Pattanayak

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 4

Résumé

Let $R$ be a prime ring with Utumi quotient ring $U$ and extended centroid $C$, $g$ a nonzero generalized derivation of $R$, $I$ a nonzero right ideal of $R$, $f(r_1,\ldots,r_k)$ a multilinear polynomial over $C$ and $n\geq 2$ be a fixed integer. If $g(f(r_1,\ldots,r_k)^n)=g(f(r_1,\ldots,r_k))^n$ for all $r_1,\ldots,r_k\in I$, then one of the following holds: (1) $I C=eRC$ for some idempotent $e\in soc (RC)$ and $f(x_1,\ldots,x_k)$ is central-valued on $eRCe$; (2) there exist $a,b\in U$ such that $g(x)=ax+xb$ for all $x\in R$ and $(a-\alpha)I=(0)$, $(b-\beta)I=(0)$ for some $\alpha,\beta\in C$ with $(\alpha+\beta)^{n-1}=1$; (3) there exists $a\in U$ such that $g(x)=ax$ for all $x\in R$ with $aI=(0)$.

Détail

Classification : 16W25, 16R50, 16N60