A theorem on derivations in semiprime rings

Let $R$ be a semiprime ring with suitably-restricted torsion, $U$ a nonzero left ideal of $R$ and $D$ : $R\rightarrow R$ a nonzero derivation. If for each $x\in U, [D(x), x]_{n} = \big[[\dots [D(x), x]\dots x], x \big]\in Z(R)$ with $n$ fixed, then $R$ must contain nonzero central ideals in case $D(U)\not= 0$.