Valuation Ring

In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D.

Given a field F, if D is a subring of F such that either x or x −1 belongs to D for every x in F, then D is said to be a valuation ring for the field F or a place of F. Since F is in this case indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. The valuation rings of a field are the maximal elements of the local subrings partially ordered by dominance, where

dominates if and .

In particular, every valuation ring is a local ring. An integral domain whose localization at any prime ideal is a valuation ring is called a Prüfer domain.