Derivations with Engel conditions on multilinear polynomials

Abstract:

Let $R$ be a prime algebra over a commutative ring $K$ with unity and let $f(X_{1}, \ldots , X_{n})$ be a multilinear polynomial over $K$. Suppose that $d$ is a nonzero derivation on $R$ such that for all $r_{1}, \ldots , r_{n}$ in some nonzero ideal $I$ of $R$, $\Big [ d\big ( f(r_{1}, \ldots , r_{n})\big ), f(r_{1}, \ldots , r_{n}) \Big ]_{k} = 0$ with $k$ fixed. Then $f(X_{1}, \ldots , X_{n})$ is central–valued on $R$ except when char $R=2$ and $R$ satisfies the standard identity $s_{4}$ in 4 variables.