A note on derivations of higher order and commutativity of prime rings

One question posed by Chung and Luh is answered. Namely, let R be a prime ring with center Z and d a non-zero derivation of R. Suppose x(dn) - x(dm) epsilon Z for all x epsilon R, where n > m are fixed non-negative integers. Then R is commutative if any one of the following conditions is satisfied: (i) m = 0; (ii) m, = 1, n > 1 an even integer, and char R not equal 2; (iii) m = 2, n > 2 an odd integer, and char R not equal 2.