Annihilators of power values of a derivation

Let R be a prime ring without nonzero nil right ideals and let U be a noncommutative Lie ideal of R. If d is a derivation of R and a is an element of R such that ad(u)(n(u)) = 0 for all u is an element of U, where n(u) is a positive integer depending on u, then either a = 0 or d = 0.