On the traces of certain classes of permuting mappings in rings

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 : R n → 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 : R n → R ${\Delta \colon R^n\rightarrow R}$ is said to be permuting n-derivation if Δ ( x 1 , x 2 , ⋯ , x i x i ' , ⋯ , x n ) = Δ ( x 1 , x 2 , ⋯ , x i , ⋯ , x n ) x i ' + x i Δ ( x 1 , x 2 , ⋯ , x i ' , ⋯ , x n ) for all x i , x i ' ∈ 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 : R n → R ${\Omega \colon R^n\rightarrow R}$ is known to be a permuting generalized n-derivation if there exists a permuting n-derivation Δ : R n → R ${\Delta \colon R^n\rightarrow R}$ such that Ω ( x 1 , x 2 , ⋯ , x i x i ' , ⋯ , x n ) = Ω ( x 1 , x 2 , ⋯ , x i , ⋯ , x n ) x i ' + x i Δ ( x 1 , x 2 , ⋯ , x i ' , ⋯ , x n ) for all x i , x i ' ∈ 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 : R n → 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.