  The group A_{m} of automophisms of a one-rooted m-ary tree admits a diagonal
monomorphism which we denote by x. Let A be an abelian state-closed (or
self-similar) subgroup of A_{m}. We prove that the combined diagonal and
tree-topological closure A* of A is additively a finitely presented Z_m
[[x]]-module where Z_m is the ring of m-adic integers. Moreover, if A* is
torsion-free then it is a finitely generated pro-m group. The group A splits
over its torsion subgroup. We study in detail the case where A* corresponds to
a cyclic Z_m[[x]]-module and when m is a prime number, we show A* to be
conjugate by a tree automorphism to one of two specific types of groups.
