The multiple holomorph of a finitely generated abelian group

W.H. Mills has determined, for a finitely generated abelian group G, the regular subgroups N≅G of S(G), the group of permutations on the set G, which have the same holomorph as G, that is, such that NS(G)(N)=NS(G)(ρ(G)), where ρ is the (right) regular representation. We give an alternative approach to Mills' result, which relies on a characterization of the regular subgroups of NS(G)(ρ(G)) in terms of commutative ring structures on G. We are led to solve, for the case of a finitely generated abelian group G, the following problem: given an abelian group (G,+), what are the commutative ring structures (G,+,⋅) such that all automorphisms of G as a group are also automorphisms of G as a ring?