Abstract
We investigate the action of derivation d in a prime ring and semiprime ring R with center Z(R) satisfying (i) (d(x circle y))(m) - (x circle y)(n) is an element of Z(R) (ii) (d(x)circle d(y))(m)=(x circle y)(n) for all x, y is an element of I, a nonzero ideal of R. Finally, as an application we apply our results to the continuous derivations on Banach algebras.