Valuation ring

Summary

A valuation ring is an integral domain $R$ with a simple property, see the definition. For each nonzero $x$ in its field of fractions, either $x$ or $1/x$ lies in $R$. Valuation rings are exactly the rings of elements of nonnegative value for a valuation. They are local rings, and their ideals are totally ordered by inclusion. Discrete valuation rings are the Noetherian valuation rings that are not fields.