Left Annihilators of Commutators with Derivation on Right Ideals

Abstract

Let R be a prime ring of characteristic different from 2, d a non-zero derivation of R, I a non-zero right ideal of R, a ∈ R, S ₄(x ₁,…, x ₄) the standard polynomial in 4 variables. Suppose that, for any x, y ∈ I, ad(x, y]), x, y]] = 0. If S ₄(I, I, I, I)I ≠ 0, then aI = ad(I) = 0.