is $\mathbb{Q}$ finitely generated as a $\mathbb{Z}_{(p)}$-algebra?

Question

I know that $\mathbb{Q}$ is not finitely generated as a $\mathbb{Z}$-algebra (and thus also not finitely generated as a $\mathbb{Z}$-module) how about $\mathbb{Q}$ as $\mathbb{Z}_{(p)}$-algebra? (or even a $\mathbb{Z}_{(p)}$-module ??)

(By $\mathbb{Z}_{(p)}$ i mean the localisation of $\mathbb{Z}$ at the prime ideal generated by some prime $p$.)

Answer 1

It is true — it is even a monogeneous algebra:

$$\mathbf Q=\mathbf Z_{(p)}[X]/(pX-1).$$

Answer 2

I just want to add that $\mathbb{Q}$ isn't finitely generated as a $\mathbb{Z}_{(p)}$-module. You can probably prove this directly but you can also do it by contradiction.

If $I = p\mathbb{Z}_{(p)} $, then $I$ is the Jacobson radical of $\mathbb{Z}_{(p)}$ and $I\mathbb{Q} = \mathbb{Q} \neq 0$. If $\mathbb{Q}$ was finitely generated, this would contradict Nakayama's Lemma.