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.
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 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).
A theorem of (Krull 1939) 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".
[a] Even further, given any totally ordered abelian group Γ, there is a valuation ring D with value group Γ (see Hahn series).
[3] It also follows from this that a valuation ring is Noetherian if and only if it is a principal ideal domain.
(By convention, a field is not a discrete valuation ring.)
A value group is called discrete if it is isomorphic to the additive group of the integers, and a valuation ring has a discrete valuation group if and only if it is a discrete valuation ring.
The units, or invertible elements, of a valuation ring are the elements x in D such that x −1 is also a member of D. The other elements of D – called nonunits – do not have an inverse in D, and they form an ideal M. This ideal is maximal among the (totally ordered) ideals of D. Since M is a maximal ideal, the quotient ring D/M is a field, called the residue field of D. In general, we say a local ring
in a field K is dominated by some valuation ring of K. Indeed, the set consisting of all subrings R of K containing A and
R is a local ring with maximal ideal containing
be a maximal extension, which clearly exists by Zorn's lemma.
By maximality, R is a local ring with maximal ideal containing the kernel of f. If S is a local ring dominating R, then S is algebraic over R; if not,
extends g; hence, S = R.) If a subring R of a field K contains a valuation ring D of K, then, by checking Definition 1, R is also a valuation ring of K. In particular, R is local and its maximal ideal contracts to some prime ideal of D, say,
, which is a valuation ring since the ideals are totally ordered.
the set of all subrings of K containing D. In particular, D is integrally closed,[8][c] and the Krull dimension of D is the number of proper subrings of K containing D. In fact, the integral closure of an integral domain A in the field of fractions K of A is the intersection of all valuation rings of K containing A.
Let X be an algebraic variety over a field k. Then we say a valuation ring R in
[10] We may describe the ideals in the valuation ring by means of its value group.
Let Γ be a totally ordered abelian group.
The set of isolated subgroups is totally ordered by inclusion.
Since the nonzero prime ideals are totally ordered and they correspond to isolated subgroups of Γ, the height of Γ is equal to the Krull dimension of the valuation ring D associated with Γ.
The most important special case is height one, which is equivalent to Γ being a subgroup of the real numbers
under addition (or equivalently, of the positive real numbers
A special case of this are the discrete valuation rings mentioned earlier.
The rational rank rr(Γ) is defined as the rank of the value group as an abelian group, A place of a field K is a ring homomorphism p from a valuation ring D of K to some field such that, for any
if and only if a prime ideal corresponding to p specializes to a prime ideal corresponding to p′ in some valuation ring (recall that if
are valuation rings of the same field, then D corresponds to a prime ideal of
If D is a valuation ring of p, then its Krull dimension is the cardinarity of the specializations other than p to p. Thus, for any place p with valuation ring D of a field K over a field k, we have: If p is a place and A is a subring of the valuation ring of p, then