Abelian group that is finitely generated as a module over the ring of integers localized at a set of primes

Definition
A group $$G$$ is termed abelian group that is finitely generated as a module over the ring of integers localized at a set of primes if it satisfies the following: there is a (possibly empty, possibly finite, possibly infinite) subset $$\pi$$ of the set of prime numbers such that $$G$$ is a finitely generated as a module over the ring $$\mathbb{Z}[\pi^{-1}]$$. Another way of putting it is that there is a finite subset $$S$$ of $$G$$ such that the $$\pi$$-powered subgroup generated by $$S$$ is the whole group $$G$$.