An Engel condition with derivation for left ideals

Abstract:

We generalize a number of results in the literature by proving the following theorem: Let $R$ be a semiprime ring, $D$ a nonzero derivation of $R$, $L$ a nonzero left ideal of $R$, and let $[x,y]=xy-yx$. If for some positive integers $t_0,t_1,\dots , t_n$, and all $x\in L$, the identity $[[\dots [[D(x^{t_0}),x^{t_1}],x^{t_2}],\dots ],x^{t_n}]=0$ holds, then either $D(L)=0$ or else the ideal of $R$ generated by $D(L)$ and $D(R)L$ is in the center of $R$. In particular, when $R$ is a prime ring, $R$ is commutative.