TITLE: Localization and p-adic completion of Integers coincide?
QUESTION [6 upvotes]: I want to know if $\mathbb{Z}_{(p)}$ (localization by a prime ideal) and $\mathbb{Z}_p$ (the completion of p-adic integers) are isomorphic. It seems true, but i don't know how to prove it. Does it holds for every PID?
Thanks.

REPLY [4 votes]: $\mathbb{Z}_{(p)}$ is a proper subring of $\mathbb{Z}_{p}$, and the latter one is complete discrete valuation ring, but $\mathbb{Z}_{(p)}$ is not complete (but still DVR with respect to the same valuation). However, if you take completion of $\mathbb{Z}_{(p)}$ with respect to the $p$-adic norm, you get $\mathbb{Z}_{p}$.