DERIVATIONS COCENTRALIZING POLYNOMIALS

Abstract

Let $R$ be a prime ring with extended centroid $C$ and $f(X_1, \ldots, X_t)$ a polynomial over $C$ which is not central-valued on $RC$. Suppose that $d$ and $\delta$ are two derivations of $R$ such that $$ d(f(x_1, \ldots, x_t))f(x_1, \ldots, x_t) -f(x_1, \ldots, x_t) \delta (f(x_1, \ldots, x_t)) \in C $$ for all $x_1, \ldots, x_t$ in $R$. Then either $d = 0 = \delta$, or $\delta = - d$ and ${f(X_1, \ldots, X_t)}^2$ is central-valued on $RC$, except when $\rm charR =2$ and $\dim_CRC = 4$.