Bulletin of the
Korean Mathematical Society BKMS

Let $K$ be a commutative ring with unity, $R$ a prime $K$-algebra of characteristic different from 2, $d$ a non-zero derivation of $R$, $I$ a non-zero right ideal of $R$, $f(x_1,\ldots,x_n)$ a multilinear polynomial in n non-commuting variables over $K$, $a\in R$. Supppose that, for any $x_1,\ldots,x_n\in I$, $a[d(f(x_1,\ldots,x_n)),f(x_1,\ldots,x_n)]=0$. If $[f(x_1,\ldots,x_n),x_{n+1}]x_{n+2}$ is not an identity for $I$ and $$S_4(I,I,I,I)I\neq 0,$$ then $aI=ad(I)=0.$

Keywords: prime rings, derivations, generalized polynomial identities

MSC numbers: Primary 16N60; Secondary 16W25