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I am studying valuation rings (beginner). I have read some theorems but still don't know a nontrivial example. Please give me an example which is not field. Also Need help to have examples of Krull dimension 1 and 2.
Many thanks.

user 1
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Sonii
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  • I hope OP is saying about Krull dimension: this link has some example which may be useful for listing some examples.https://en.wikipedia.org/wiki/Krull_dimension – Germain Apr 14 '16 at 14:19
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    The simplest valuation rings (and most basic) are discrete valuation rings. These are all of Krull dimension one and ubiquitous in Number theory and Algebra. Most number theory books will discuss $p$-adic valuations for primes $p$. – Mohan Apr 14 '16 at 16:38
  • You can try starting with polynomial rings. Do you know what the valuation on $k[x]$ is? Do you know about localization? – John Martin Apr 14 '16 at 18:02
  • No, john, can it be a part of answer or is an other question that i should ask? – Sonii Apr 16 '16 at 16:35

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As I dont know how much do you know about valuation rings and which book you are reading I can only say 2 points; waiting your response:

  • K[x] is PID (has Krull dimension 1). Localize it to the maximal ideal (x). It is a valuation ring of Krull dimension 1.
  • For a valuation ring of Krull dimension 2, see mr.bigproblem's answer to this question.
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