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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. –  Arturo Magidin Feb 16 '11 at 5:22
And can you make this community wiki, please? –  awllower Feb 16 '11 at 6:52
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?) –  Akhil Mathew Feb 16 '11 at 7:19
Sorry, I get your point now, I will remove the tag. –  awllower Feb 16 '11 at 9:35
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. –  awllower Feb 17 '11 at 0:09

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