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 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

For example, consider the inclusion ${\displaystyle (\mathbb {C} [[X^{2}]],(X^{2}))\hookrightarrow (\mathbb {C} [[X]],(X))}$. These are all subrings in the field of bounded-below power series ${\displaystyle \mathbb {C} ((X))}$. Every local ring in a field K is dominated by some valuation ring of K.

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