What is the spectrum of $\mathbb{Z}_{\langle 5 \rangle}$?

Question

I'm trying to describe the spectrum of this ring. I've figured that this is a local ring with maximal ideal generated by $\frac{5}{1}$, so I have that $\operatorname{Spec}(\mathbb{Z}_{\langle 5\rangle})=\{\langle5\rangle,\dots\}$. I'm not sure how to figure out the other elements of the spectrum... how do I go about it?

Answer 1

Primes of the localization $\mathbb{Z}_{(5)}$ correspond bijectively to primes of $\mathbb{Z}$ contained within $(5)$. (More generally, primes in $S^{-1} R$ correspond to primes of $R$ contained within $R \backslash S$.) The bijection sends an ideal $\mathfrak{p} \subseteq S^{-1}R$ to its preimage $f^{-1}(\mathfrak{p})$ under the natural map $f : R \rightarrow S^{-1} R.$

So in particular the primes of $\mathbb{Z}_{(5)}$ are $(5)$ and $(0)$.