On the range of a derivation

Let ?? be a Banach algebra and ??: ?? ?? ?? a derivation. In this paper, it is proved, under certain conditions, that ???????? ?? ????????????, where ???????????? is the Jacobson radical of ??. Moreover, we prove that if ?? is unital and ??:?? ?? ?? is a continuous derivation, then ???????? ?? ??????????????? ?? ?? ??????????????? ?? ?? ????????? ????, where ?????????? denotes the set of all primitive ideals such that ?? ?? is commutative, ?????????? denotes the set of all maximal (modular) ideals such that ?? ?? is commutative, and ??? is the set of all non-zero multiplicative linear functionals from ?? into ??. In addition, we present several results about the range of a derivation on algebras having the property (??).