Definition 10.50.1. Valuation rings.

1. Let $K$ be a field. Let $A$, $B$ be local rings contained in $K$. We say that $B$ dominates $A$ if $A \subset B$ and $\mathfrak m_ A = A \cap \mathfrak m_ B$.

2. Let $A$ be a ring. We say $A$ is a valuation ring if $A$ is a local domain and if $A$ is maximal for the relation of domination among local rings contained in the fraction field of $A$.

3. Let $A$ be a valuation ring with fraction field $K$. If $R \subset K$ is a subring of $K$, then we say $A$ is centered on $R$ if $R \subset A$.

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