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As we all know, the ring $Z_p$ can be constructed as the projective limit of the rings $Z/p^{n}Z$.
Now is there any generalization such as the p-adic completions of a Dedekind Domain?
This might be said to be inspired by the general treatment of extensions of Dedekind Domains from the treatment of algebraic number fields.
In any case, thanks very much.

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    Yes; you can do valuations with any prime ideal in the exactly analogous manner, and do the corresponding limit (projective limit) which yields precisely the completion relative to the valuation determined by the prime ideal.2011-02-16
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    And can you make this community wiki, please?2011-02-16
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    Dear @awillower, I can wikify your question, but why? It is not a soft question. Why should people who post good answers not get reputation from it? (Or why should your asking a technical question not earn you reputation?)2011-02-16
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    Sorry, I get your point now, I will remove the tag.2011-02-16
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    I have found a reasonable explanation such that this can be done:As *Emil Artin* once said, the **product formula** is an important characterization of such fields, and since in the Dedekind case, the product formula isn't changed, this is imaginable.2011-02-17
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    In fact, at the beginning, I viewed this question as pretty abnormal, while now I realized that this question clarified the importance of Dedekind domains, and emphasized one of the proofs for the product formula which is the only one I know, i.e. via. unique factorization of prime ideals.2011-02-17

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