Derivations of rings and Banach algebras

Abstract

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