Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

中文翻译：

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半素环上的加性 n 对易映射

让R是一个质心延长的半素环C和问的商的最大右环R. 放 [是的,X]1= [是的,X] =yx-xy为了X,是的∈问和归纳[是的,X]ķ= [[是的,X]ķ-1,X] 为了ķ> 1. 假设F：R→问是一个满足 [F(X),X]n= 0 全部X∈R， 在哪里n是一个固定的正整数。则可以证明存在 λ ∈C和一个加法图μ：R→C这样F(X) = λX+μ(X） 对所有人X∈R. 这对 Brešar 在 1996 年提出的此类功能性身份的未解决问题给出了肯定的答案。