valuation ring

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valuation ring

TFAE:

  • \(A\) is a valuation ring
  • \(A\) is an integral domain \(A\) with such that for every \(x\in \operatorname{ff}(A)\), either \(x\in A\) or \(x^{-1}\in A\).
  • The ideal poset \(\operatorname{Id}(A)\) is totally ordered.
  • The divisibility poset of \(A\) is totally ordered.
  • There is a totally ordered value group \(G\) and a valuation \(v: \operatorname{ff}(A) \to G\cup\left\{{\infty}\right\}\) such that \begin{align*} A = \left\{{x\in \operatorname{ff}(A) {~\mathrel{\Big\vert}~}v(x) \geq 0}\right\} = \left\{{x\in \operatorname{ff}(A) {~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert}_v \leq 1}\right\} .\end{align*}

Facts

  • Valuation rings are integrally closed.
  • A DVR is an integral domain \(R\) that arises as the valuation ring of \(\operatorname{ff}(R)\) with respect to a discrete valuation.
  • The integral closure of an integral domain \(A\) in \(\operatorname{ff}(A)\) is the intersection of all valuation rings of \(\operatorname{ff}(A)\) containing \(A\).
#NT/algebraic #todo/add-references