Let R be a noncommutative semi prime ring. Suppose that there exists a derivation d : R $ o$ R such that for all x $in$ R, either [[d(x),x], d(x)] = 0 or $langle$$langle(x),;x angle,;d(x) angle$ = 0. In this case [d(x), x] is nilpotent for all x $in$ R. We also apply the above results to a Banach algebra theory.