What is maximal ideal $\mathbb{Z}\_{(p)}$ ? And how it is calculated?
Also why $\mathbb{Z}\_{(p)}/p\mathbb{Z}\_{(p)} = \mathbb{F}\_p.$?

I know $p\mathbb{Z}\_{(p)}$ is maximal ideal for $\mathbb{Z}\_{(p)}$ but how? I know $\mathbb{Z}\_{(p)} $ is local so it has unique maximal ideal .