Derivations in prime rings

We prove two theorems that are easily conjectured, namely: (1) In a prime ring of characteristics not 2, if the iterate of two derivations is a derivation, then one of them is zero; (2) If d is a derivation of a prime ring such that, for all elements a of the ring, ad(a) -d(a)a is central, then either the ring is commutative or d is zero. DEFINITION. A ring R is called prime if and only if xay= 0 for all aER implies x=O or y=O. From this definition it follows that no nonzero element of the centroid has nonzero kernel, so that we can divide by the prime p, unless px = 0 for all x in R, in which case we call R of characteristic p. A known result that will be often used throughout this paper is given in