Vanishing Power Values of Commutators with Derivations on Prime Rings

Abstract

Let R be a prime ring of char R≠2, d a nonzero derivation of R and ρ a nonzero right ideal of R such that [[d(x),x]n,[y,d(y)]m]t=0 for all x,y∈ρ, where n≥0, m≥0, t≥1 are fixed integers. If [ρ,ρ]ρ≠0, then d(ρ)ρ=0.