The Stacks project

Definition 10.50.13. Let $A$ be a valuation ring.

  1. The totally ordered abelian group $(\Gamma , \geq )$ of Lemma 10.50.12 is called the value group of the valuation ring $A$.

  2. The map $v : A - \{ 0\} \to \Gamma $ and also $v : K^* \to \Gamma $ is called the valuation associated to $A$.

  3. The valuation ring $A$ is called a discrete valuation ring if $\Gamma \cong \mathbf{Z}$.

Comments (1)

Comment #8751 by Anonymous on

Pedantic point, but do you want to allow fields to be discrete valuation rings? For example, Tag 034X seems to consider fields as discrete valuation rings, but the value group will trivial.

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  • 3 comment(s) on Section 10.50: Valuation rings

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