Hi, I am looking for examples of non-discrete valuation rings. Could you help me? Thanks
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7$\begingroup$ Please read the FAQ; this question does not seem to be appropriate for MO. Anyway I give you a hint: Open a book which treats valuation rings, there are many examples. $\endgroup$– Martin BrandenburgFeb 2, 2011 at 17:28
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7$\begingroup$ Here is a good example: Let $k$ be a field and consider the field of formal power series $\sum_{i \in I} a_i t^i$ where $I$ can be any well-ordered en.wikipedia.org/wiki/Well_ordered subset of $\mathbb{R}$. Sum and product are defined formally by $\sum (a_i+b_i) t^i$ and $\sum a_i b_j t^{i+j}$. Exercise: The sum or product of two formal power series of this form is another one, and every coefficient of the sum or product is a finite polynomial in the coefficents of the summands/multiplicands. The valuation is $v(\sum a_i t^i) = \min(i : a_i \neq 0)$. $\endgroup$– David E SpeyerFeb 2, 2011 at 18:10
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2 Answers
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I suggest having a look to Bosch, Güntze and Remmert's "Non-archimedean analysis: a systematic approach to rigid analytic geometry" (1984)
They cover quite a lot of things about valuations and norms.
answered Feb 2, 2011 at 18:11
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Take any valuation ring, which is not noetherian.
