Apologies in advance if this is a vague question. So I am trying to calculate the $\mathbf{Z}$-rank of a finitely generated torsion free module $M$ (I won't go into the details of what this module $M$ actually is since it would take far too long. Suffice to say, it is monogenic). Now as I understand it,
$dim_\mathbf{Z}(M)=dim_\mathbf{Q}(M_\mathbf{Q})$,
where $M_\mathbf{Q}$ is the localization of $M$ in the field of fractions of $\mathbf{Z}$. As I am not especially adept at these localization arguments, I was wondering if there are any straightforward methods I could adapt to calculate this dimension, or if there are any online resources that explain this fully?
If $M$ is isomorphic to a direct sum of $n$ copies of $\mathbb{Z}$ and some finite abelian group, then the Z-rank is then $n$.
$\mathbb{Z}$-modules are simply abelian groups. If it is finitely generated then it must be isomorphic to $\mathbb{Z}^{\bigoplus n}\oplus\mathbb{Z}_{m_1}\oplus\cdots\oplus\mathbb{Z}_{m_k}$. After localizing to the field of fraction $\mathbb{Q}$, only the $\mathbb{Z}$-summands survive (reason: for $k\otimes r\in\mathbb{Z}_m\otimes_{\mathbb{Z}}\mathbb{Q}$, $k\otimes r=k\otimes m\cdot\frac{r}{m}=mk\otimes\frac{r}{m}=0$). You will get $\mathbb{Q}^{\bigoplus n}$. Thus the rank is indeed $n$.