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 $\Gamma \cong \mathbf{Z}$.

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