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

It is true — it is even a monogeneous algebra:

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