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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 nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is 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. In particular, every valuation ring is a local ring.
The valuation rings of a field are the maximal elements of the set of the local subrings in the field partially ordered by dominance or refinement, where
Every local ring in a field K is dominated by some valuation ring of K.
An integral domain whose localization at any prime ideal is a valuation ring is called a Prüfer domain.