Finite index and finite codimension

An old idea of M. Hall on finitely generated subgroups of free groups is developed. We show that it implies that such subgroups have "roots" which are normalizers of certain other subgroups. Similarly in free algebras or group rings of free groups over a field every finitely generated right ideal has a root, which is the unique maximal subalgebra that contains the ideal as an ideal of finite codimension. In analogy to the group case, it is an "idealizer" of another, related, ideal. We also define the "Hall index" of a subgroup of a free group and relate it to Howson's theorem.