# Localization and p-adic completion of Integers coincide?

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.

• $\mathbb{Z}_{(p)}$ is dense in $\mathbb{Z}_p$ and in both $(p^n)$ are the only ideals – reuns Nov 24 '18 at 2:40

## 2 Answers

No, $$\mathbb{Z}_p$$ is much larger than $$\mathbb{Z}_{(p)}$$. Indeed, $$\mathbb{Z}_p$$ is uncountable, since it has an element $$\sum a_np^n$$ for any sequence of coefficients $$a_n\in\{0,1,\dots,p-1\}$$. On the other hand, $$\mathbb{Z}_{(p)}$$ is a subring of $$\mathbb{Q}$$ (the rationals with denominator not divisible by $$p$$), so it is countable.

$$\mathbb{Z}_{(p)}$$ 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 $$\mathbb{Z}_{p}$$.