Abstract
In the present paper it is shown that if R is a semiprime ring of characteristic different from two which admits a derivation d such that [d(x(m)), d(y(n))] = +/-[x(m), y(n)] for all x, y is an element of R, where m >= 1, n >= 1 are fixed positive integers, then R is commutative. Further using this result it is established that if A is a semiprime Banach algebra and H-1 and H-2 are nonvoid open subsets of A which admits a continuous derivation d : U -> U such that [d(x(m)), d(y(n))] +/- [x(m), y(n)] = 0 for all x. H-1 and y. H-2, where m, n are no longer fixed rather they depend on the pair of elements x, y is an element of U, then A is commutative.