A Note on Power Values of Derivation in Prime and Semiprime Rings

Let r be a ring with derivation d, such that (d(xy))^n = (d(x))^n(d(y))^n for all x, y є r and n ≥ 1 a fixed integer. in this paper, we show that if r is prime, then d = 0 or r is commutative. if r is semiprime, then d maps r into its center. moreover in semiprime case let a = o(r) be the orthogonal completion of r and b = b(c) be the boolian ring of c, where c is the extended centroid of r. then there exists an idempotent e є b such that ea is a commutative ring and d induces a zero derivation on (1 − e)a.