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Valuation ring questions

  1. Dec 8, 2008 #1
    Let [tex]K[/tex] be a field, [tex]\nu : K^* \rightarrow \texbb{Z}[/tex] a discrete valuation on [tex]K[/tex], and [tex]R=\{x \in K^* : \nu(x) \geq 0 \} \cup \{0\}[/tex] the valuation ring of [tex]\nu[/tex]. For each integer [tex]k \geq 0[/tex], define [tex]A_k=\{r \in R : \nu(r) \geq k \} \cup \{0\}[/tex].

    (a) Prove that for any [tex]k[/tex], [tex]A_k[/tex] is a principal ideal, and that [tex]A_0 \supseteq A_1 \supseteq A_2 \supseteq\ldots[/tex]
    (b) Prove that if [tex]I[/tex] is any nonzero ideal of [tex]R[/tex], then [tex]I=A_k[/tex] for some [tex]k \geq 0[/tex].
  2. jcsd
  3. Dec 9, 2008 #2


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    Science Advisor
    Homework Helper

    the only tools you have are facts about integers (well ordering) and the definition of a valuation. try looking at an element of minimal valuation.
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