Symmetric skew 4-derivations on prime rings

Abstract

For a ring R with an automorphism α a 4-additive mapping D : R4−→ R is called a skew 4-derivation w.r.t. α if it is a α-derivation of R for each argument. Namely it is always an α-derivation of R for the argument being left once (3) arguments are fixed by (3) elements in R. In the present note, begin with a result of Jung and Park [5], we prove that if a skew 4-derivation D associated with an automorphism α with trace f of a noncommutative prime ring R under suitable torsion condition satisfying [f(x),α(x)] = 0 for all x ∈ I, a nonzero ideal of R, then D = 0.