TRACES OF PERMUTING GENERALIZED N-DERIVATIONS OF RINGS.

Let n ≥ 1 be a fixed positive integer and R be a ring. A permuting n-additive map Ω : Rn → R is known to be permuting generalized n-derivation if there exists a permuting n-derivation Δ : Rn → R such that Ω(x1, x2, ⋯,xixi⋯,xn) = Ω(x1, x2, ⋯,xi,⋯xn)xi + xi Δ(x1, x2, ⋯,xi⋯xn) holds for all xi, xi ∈ R. A mapping δ : R → R defined by δ(x) = Δ(x, x, ⋯, x) for all x ∈ R is said to be the trace of Δ. The trace &# of can be defined in ω of Ω 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 Δ such that the trace δ of Δ satisfies [[δ(x), x], x] ∈ Z (R) for all x ∈ R, then ͐ is commuting on R. Besides other related results it is also shown that in a n-torsion free prime ring if the trace ω of a permuting generalized n-derivation is centralizing on R, then ω is commuting on R. [ABSTRACT FROM AUTHOR]