Abstract
Let R be a semiprime ring with center Z and extended centroid C . For a fixed integer n ≥ 2, the trace δ:R→R${\delta \colon R\rightarrow R}$ of a permuting n -additive mapping D:Rn→R${D\colon R^n\rightarrow R}$ is defined as δ(x)=D(x,...,x)${\delta (x)=D(x,\ldots ,x)}$ for all x ∈ R . The notion of permuting n -derivation was introduced by Park [J. Chungcheong Math. Soc. 22 (2009), no.3, 451–458] as follows: a permuting n -additive mapping Δ:Rn→R${\Delta \colon R^n\rightarrow R}$ is said to be permuting n -derivation if Δ(x1,x2,⋯,xixi',⋯,xn)=Δ(x1,x2,⋯,xi,⋯,xn)xi'+xiΔ(x1,x2,⋯,xi',⋯,xn)forallxi,xi'∈R.$ \Delta (x_1,x_2,\dots , x_ix_i^{\prime },\dots , x_n)=\Delta (x_1,x_2,\dots , x_i,\dots , x_n)x_i^{\prime }+ x_i\Delta (x_1,x_2,\dots , x_i^{\prime },\dots , x_n)\quad \text{for all }x_i ,x_i^{\prime } \in R. $ A permuting n -additive mapping Ω:Rn→R${\Omega \colon R^n\rightarrow R}$ is known to be a permuting generalized n -derivation if there exists a permuting n -derivation Δ:Rn→R${\Delta \colon R^n\rightarrow R}$ such that Ω(x1,x2,⋯,xixi',⋯,xn)=Ω(x1,x2,⋯,xi,⋯,xn)xi'+xiΔ(x1,x2,⋯,xi',⋯,xn)forallxi,xi'∈R.$ \Omega (x_1,x_2,\dots , x_ix_i^{\prime },\dots , x_n)=\Omega (x_1,x_2,\dots , x_i,\dots , x_n)x_i^{\prime }+ x_i\Delta (x_1,x_2,\dots , x_i^{\prime },\dots , x_n)\quad \text{for all }x_i ,x_i^{\prime } \in R. $ The main result of this paper states that if I is a nonzero ideal of a semiprime ring R and Δ:Rn→R${\Delta :R^n\rightarrow R}$ is a permuting n -derivation such that Δ(I,...,I)≠{0}${\Delta (I,\ldots ,I)\ne \lbrace 0\rbrace }$ and [δ(x),x]=0${[\delta (x),x]=0}$ for all x ∈ I , where δ is the trace of Δ, then R contains a nonzero central ideal. Furthermore, some related results are also proven.