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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.

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

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

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