localizations of a direct sum module

Question

Consider the $\mathbb{Z}$-module $M=\bigoplus{\mathbb{Z}/p\mathbb{Z}}$, where the direct sum is taken over the set of all prime numbers. How do I show that the localizations $M_\mathfrak{p}$ are finitely generated $\mathbb{Z}_\mathfrak{p}$-modules for any prime ideal $\mathfrak{p}$ of $\mathbb{Z}$?

Answer 1

Recall the following facts:

(1) Direct limits, direct sums and tensor products commute up to isomorphism

(2) The localization of a $R$-module $M$ at a prime $\mathfrak{p}$ is isomorphic to the module obtained by extending $M$ to scalars in $R_{\mathfrak{p}}$

(3) $(\mathbb{Z}/p\mathbb{Z}) \otimes_{\mathbb{Z}}\mathbb{Z}_q$ is isomoprhic to $\mathbb{Z}/p\mathbb{Z}$ if $p=q$ and $0$ otherwise.

From these three facts, it follows that

$$M_{\mathfrak{p}} = M \otimes_{\mathbb{Z}} \mathbb{Z}_{\mathfrak{p}} = (\displaystyle\lim_{\rightarrow} \text{ } \displaystyle \bigoplus_{p<n} \text{ } \mathbb{Z}/p\mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{Z}_{\mathfrak{p}} \cong \displaystyle\lim_{\rightarrow} \text{ } \displaystyle \bigoplus_{p<n} \text{ } (\mathbb{Z}/p\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}_{\mathfrak{p}}) \cong \mathbb{Z}/\mathfrak{p}$$

and your claim follows.