On nilpotency of the separating ideal of a derivation

We prove that the separating ideal S(D) of any derivation D on a commutative unital algebra B is nilpotent if and only if S(D) n (n Rn) is a nil ideal, where R is the Jacobson radical of B . Also we show that any derivation D on a commutative unital semiprime Banach algebra B is continuous if and only if n(S(D))n = {O} . Further we show that the set of all nilpotent elements of S(D) is equal to n(S(D)nP) , where the intersection runs over all nonclosed prime ideals of B not containing S(D) . As a consequence, we show that if a commutative unital Banach algebra has only countably many nonclosed prime ideals then the separating ideal of a derivation is nilpotent.