On traces of permuting n-derivations on prime ideals

. In this article we investigate some properties of permuting n-derivations acting on a prime ideal. More precisely, let n ≥ 2 be a fixed positive integer, P be a prime ideal of a ring R such that R / P is ( n + 1 ) !-torsion free. If there exists a permuting n -derivation ∆ : R n −→ R such that the trace δ of ∆ satisfies (cid:2) [ δ ( x ) , x ] , x (cid:3) ∈ Z ( R / P ) for all x ∈ R , then ∆ ( R n ) ⊆ P or R/P is commutative.