Localization of module of noetherian ring is f.g. but module isn't f.g.

Question

Give an example of $A$-module $M$, where $A$ is a noetherian ring that $M$ is not finitely generated, but $M_{\mathfrak{p}}$ for all $\mathfrak{p} \in Spec(A)$ is finitely generated.

I don't have any idea...

Answer 1

Let $A$ be any noetherian ring with infinitely many maximal ideals (For example, the integers) and $$M = \bigoplus_{\mathfrak m \subset A \text{ maximal}} A/\mathfrak m.$$

To compute localizations of $M$, note that localization commutes with arbitrary direct sums and we have $\operatorname{Supp} A/\mathfrak m = \{\mathfrak m\}$.