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Concept
Valuation ring
Summary
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
dominates if and .
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.
There are several equivalent definitions of valuation ring (see below for the characterization in terms of dominance). For an integral domain D and its field of fractions K, the following are equivalent:
For every nonzero x in K, either x is in D or x−1 is in D.
The ideals of D are totally ordered by inclusion.
The principal ideals of D are totally ordered by inclusion (i.e. the elements in D are, up to units, totally ordered by divisibility.)
There is a totally ordered abelian group Γ (called the value group) and a valuation ν: K → Γ ∪ {∞} with D = { x ∈ K | ν(x) ≥ 0 }.
The equivalence of the first three definitions follows easily. A theorem of states that any ring satisfying the first three conditions satisfies the fourth: take Γ to be the quotient K×/D× of the unit group of K by the unit group of D, and take ν to be the natural projection. We can turn Γ into a totally ordered group by declaring the residue classes of elements of D as "positive".
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