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

The totally ordered abelian group $(\Gamma , \geq )$ of Lemma 10.50.12 is called the

*value group*of the valuation ring $A$.The map $v : A - \{ 0\} \to \Gamma $ and also $v : K^* \to \Gamma $ is called the

*valuation*associated to $A$.The valuation ring $A$ is called a

*discrete valuation ring*if $\Gamma \cong \mathbf{Z}$.

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