Valuation ring

In abstract algebra, a valuation ring is an integral domain D such that for every non-zero 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,^[1] where

${\displaystyle (A,{\mathfrak {m}}_{A})}$ dominates ${\displaystyle (B,{\mathfrak {m}}_{B})}$ if ${\displaystyle A\supseteq B}$ and ${\displaystyle {\mathfrak {m}}_{A}\cap B={\mathfrak {m}}_{B}}$.^[2]

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.