Definition 10.50.1. Valuation rings.

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 $\mathfrak m_ A = A \cap \mathfrak m_ B$.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 $A$.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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