Characterization finitely generated $\mathbb{Z}$-module of $\mathbb{Q}$

Question

It is well-known that $\mathbb{Q}$ is not finitely generated as $\mathbb{Z}$-module. Moreover, if we consider $\mathbb{Z}_{(2)}$, it is a proper $\mathbb{Z}$-submodule of $\mathbb{Q}$ which is not finitely generated.

Well, my question is: there exists a characterization of finitely generated $\mathbb{Z}$-submodules of $\mathbb{Q}$?

Answer 1

Yes: they're exactly the cyclic submodules, because if you call $m=\prod_{j=1}^k m_j$, then $$\Bbb Z\frac{n_1}{m_1}+\cdots+\Bbb Z\frac{n_k}{m_k}=\Bbb Z\frac{\operatorname{gcd}\left(\frac m{m_1}n_1,\cdots, \frac{m}{m_k}n_k\right)}{m}$$

by Bézout's theorem.