Let R be a prime ring with extended centroid C and maximal right ring of quotients Q. Set ${[y,x]}_1=[y,x]=yx-xy$ for $x,y\in Q$ and inductively ${[y,x]}_k=[{[y,x]}_{k-1},x]$ for all integers $k\ge 2$. Suppose that L is a noncentral Lie ideal of R and $f:L\to Q$ is an additive map satisfying ${[f(x),x]}_n=0$ for all $x\in L$, where n is a fixed positive integer. It is shown that there exist $\lambda \in C$ and an additive map $\mu :L\to C$ such that $f(x)=\lambda x+\mu (x)$ for all $x\in L$ except when $\mathrm{char}R=2$ and $R\subseteq {\mathrm{M}}_2(F)$, the $2\times 2$ matrix ring over a field F. This result naturally generalizes the classical theorem for derivations by Lanski and the recent theorem for X-generalized skew derivations by De Filippis and Wei. Moreover, it gives an affirmative answer to the unsolved problem of such functional identities raised by Beidar in 1998.

令R是质心扩展C和商Q 的最大右环的素环。放${[是,X]}_1=[是,X]=是X-X是$ 为了 $X,是\in 问$ 并归纳地 ${[是,X]}_{克}=[{[是,X]}_{克-1},X]$ 对于所有整数 $克\ge 2$. 假设L是R的非中心李理想，并且$F：升\to 问$ 是一个加法映射满足 ${[F(X),X]}_n=0$ 对所有人 $X\in 升$，其中n是一个固定的正整数。表明存在$\lambda \in C$ 和附加图 $\mu ：升\to C$ 以至于 $F(X)=\lambda X+\mu (X)$ 对所有人 $X\in 升$ 除了当 $\mathrm{字符}\mathrm{电阻}=2$ 和 $\mathrm{电阻}\subseteq {\mathrm{米}}_2(F)$， 这 $2\times 2$场F 上的矩阵环。这个结果自然地推广了 Lanski 推导的经典定理和De Filippis 和 Wei 的X广义偏斜推导的最新定理。并且，它对北达在1998年提出的这种功能身份的未解决问题给出了肯定的回答。