---- > [!definition] Definition. ([[valuation ring]]) > A **valuation ring** is an [[integral domain]] $D$ satisfying the property that for any nonzero $x \in \text{Frac }D$,[^1] at least one of $x$ or $x ^{-1}$ in fact belongs to $D$. > > > A **valuation ring of a [[field]] $k$** is an [[integral domain]] $R$ satisfying $\text{Frac }R=k$ and for all $x \in k$, either $x \in R$ or $x ^{-1} \in R$. > Connect to [[discrete valuation]]. ---- #### [^1]: $\text{Frac }D$ denotes the [[field of fractions]] of $D$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```