I know that $\mathbb{Z}_{(p)}$ is a local ring because it's the localization of $\mathbb{Z}$ over $p$, but is there a direct way to prove that and find its unique maximal ideal?
I've been thinking over this for a while but nothing in the right direction comes to my mind.
Every element $a/b$ of that ring (in which $b$ is not divisible by $p$, of course) such that $a$ is not divisible by $p$ is clearly invertible. It follows that the complement of the ideal generated by $p/1$ is entirely composed of invertible elements. This is enough to conclude that the ring is local and that that ideal is the maximal ideal.