I'm trying to find the spectrum of $\mathbb{Z}[\frac{1}{6}]$. For that, I wanted to look at it as a Localization of $A := \mathbb{Z}$ with respect to $S:=\{6\}$. Then $\mathbb{Z}[\frac{1}{6}] \cong S^{-1}A$.
I know that if $\mathfrak{p} \in spec\{A\}$ then $A_{\mathfrak{p}}$ is a localization at $\mathfrak{p}$. But does it even tell me something about the spectrum?
Can someone tell me how one can calculate the spectrum of a localization?
Thanks in advance!
If $A$ is a ring, $S$ a multiplicative subset, and $S^{-1}A$ is the localization, then there is a bijection between the prime ideals of $S^{-1}A$ and the prime ideals of $A$ which don't intersect $S$.
Applying this to our case, the prime ideals of $\Bbb Z[\frac16]$ are in bijection with prime ideals of $\Bbb Z$ which don't contain $6$.