Let $ngeq 1$ be a fixed positive integer and $R$ be a ring.A permuting $n$-additive map $Omega:R^no R$ is known to bepermuting generalized $n$-derivation if there exists a permuting $n$-derivation $Delta:R^no R$such that $Omega(x_1,x_2,cdots, x_ix_i^{'},cdots, x_n)=Omega(x_1,x_2,cdots, x_i,cdots, x_n)x_i^{'}+ x_iDelta(x_1,x_2,cdots, x_i^{'},cdots, x_n)$ holds for all $x_i ,x_i^{'} in R$. A mapping $delta:R o R$ defined by $delta(x)=Delta(x,x,cdots,x)$ for all $xin R$ is said to be the trace of $Delta$.The trace $omega$ of $Omega$ can be defined in the similar way.The main result of the present paper states that if $R$ is a $(n+1)!$-torsion free semi-prime ring which admits a permuting $n$-derivation$Delta$ such that the trace $delta$ of $Delta$ satisfies $[[delta(x),x],x]in Z(R)$ for all $xin R,$ then$delta$ is commuting on $R$. Besides other related results it is also shown that in a $n!$-torsion free prime ring if the trace $omega$of a permuting generalized $n$-derivation $Omega$ is centralizing on $R,$ then $omega$ is commuting on $R$.
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机译：假设$ n geq 1 $是一个固定的正整数，而$ R $是一个环。一个已知的置换$ n $可加映射$ Omega：R ^ n 到R $可以置换广义$ n $-导数存在一个置换$ n $-导数$ Delta：R ^ n 至R $使得$ Omega（x_1，x_2， cdots，x_ix_i ^ {'}， cdots，x_n）= Omega（x_1， x_2， cdots，x_i， cdots，x_n）x_i ^ {'} + x_i Delta（x_1，x_2， cdots，x_i ^ {'}， cdots，x_n）$为所有$ x_i，x_i ^ { '} in R $。由$ delta（x）= Delta（x，x， cdots，x）$为R $中的所有$ x定义的$ delta：R 到R $的映射被称为$ 的踪迹可以用类似的方式定义$ Omega $的痕迹$ omega $。本文的主要结果表明，如果$ R $是$（n + 1）！素环，它允许置换$ n $-导数$ Delta $，使得$ Delta $的轨迹$ delta $满足Z [R] $中的$ [[ delta（x），x]，x] 对于R中所有$ x ，$然后$ delta $在$ R $上进行通勤。除其他相关结果外，还显示出，在一个无扭曲的素数环中，如果置换的广义n导数ω的轨迹ω集中在$ R上，则$ omega $在$ R $上下班。
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