AN IDENTITY WITH DERIVATIONS ON RINGS AND BANACH ALGEBRAS

The main purpose of this paper is to study the following: Let m, n, and i-1, 2,...,n, be positive integers and let R be a 2m(m+k(1)+k(2)+...+k(n))!-torsion free semiprime ring. Suppose that there exist derivations D-i, : R R, i = 1,2,...,n + 1, such that Di (x(m))x(ki) + x(ki) D-2 (x(m) + +K-n +...+ x(k1)D(n)+l (x(m))=0 holds for all x epsilon R. Then we prove that D-1 + D-2 +...+ Dn+1 = 0 and that the derivation k(1)D(2) (k(1) + k(2))D-3 +...+(k(1) + k(2) )D-3 +...+ maps R into its center. We also obtain a range inclusion result of continuous derivations on Banach algebras.