POWER CENTRAL VALUES OF DERIVATIONS ON MULTILINEAR POLYNOMIALS

Abstract

Let $R$ be a prime ring with extended centroid $C$, $\rho$ a nonzero right ideal of $R$, $d$ a nonzero derivation of $R$, $f(X_1, \ldots, X_t)$ a multilinear polynomial over $C$, $a\in R$ and $n$ a fixed positive integer.

[(I)] If $ad(f(x_1, \ldots, x_t))^n=0$ ($d(f(x_1, \ldots, x_t))^na=0$) for all $x_1, \ldots, x_t$ $\in\rho$, then either $a\rho=0$ ($a=0$ resp.), $d(\rho)\rho=0$ or $\rho C = eRC$ for some idempotent $e$ in the socle of $RC$ such that $f(X_1, \ldots, X_t)$ is central-valued on $eRCe$.

[(II)] If $ad(f(x_1, \ldots, x_t))^n\in C$ ($d(f(x_1, \ldots, x_t))^na\in C$) for all $x_1, \ldots, x_t$ $\in\rho$ and $ad(f(y_1,\ldots,y_t))^n\ne 0$ $(d(f(y_1,\ldots,y_t))^na\ne 0)$ for some $y_1,\ldots,y_t\in\rho$, then either $f(\rho)\rho=0$ or $f(X_1, \ldots, X_t)$ is central-valued on $RC$ unless dim$_CRC=4$.