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
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.
Definitions.
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:
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".
Even further, given any totally ordered abelian group Γ, there is a valuation ring D with value group Γ (see Hahn series).
From the fact that the ideals of a valuation ring are totally ordered, one can conclude that a valuation ring is a local domain, and that every finitely generated ideal of a valuation ring is principal (i.e., a valuation ring is a Bézout domain). In fact, it is a theorem of Krull that an integral domain is a valuation ring if and only if it is a local Bézout domain. It also follows from this that a valuation ring is Noetherian if and only if it is a principal ideal domain. In this case, it is either a field or it has exactly one non-zero prime ideal; in the latter case it is called a discrete valuation ring. (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.
Very rarely, "valuation ring" may refer to a ring that satisfies the second or third condition but is not necessarily a domain. A more common term for this type of ring is "uniserial ring".
Dominance and integral closure.
The units, or invertible elements, of a valuation ring are the elements "x" such that "x" −1 is also a member of D. The other elements of "D", called nonunits, do not have an inverse, 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 formula_31 dominates a local ring formula_32 if formula_33 and formula_34; in other words, the inclusion formula_35 is a local ring homomorphism. Every local ring formula_36 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 formula_37 is nonempty and is inductive; thus, has a maximal element formula_38 by Zorn's lemma. We claim "R" is a valuation ring. "R" is a local ring with maximal ideal containing formula_39 by maximality. Again by maximality it is also integrally closed. Now, if formula_40, then, by maximality, formula_41 and thus we can write:
Since formula_43 is a unit element, this implies that formula_44 is integral over "R"; thus is in "R". This proves "R" is a valuation ring. ("R" dominates "A" since its maximal ideal contains formula_45 by construction.)
A local ring "R" in a field "K" is a valuation ring if and only if it is a maximal element of the set of all local rings contained in "K" partially ordered by dominance. This easily follows from the above.
Let "A" be a subring of a field "K" and formula_46 a ring homomorphism into an algebraically closed field "k". Then "f" extends to a ring homomorphism formula_47, "D" some valuation ring of "K" containing "A". (Proof: Let formula_48 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, formula_49 contains a polynomial ring formula_50 to which "g" extends, a contradiction to maximality. It follows formula_51 is an algebraic field extension of formula_52. Thus, formula_53 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, formula_45. Then formula_55 since formula_38 dominates formula_57, which is a valuation ring since the ideals are totally ordered. This observation is subsumed to the following: there is a bijective correspondence formula_58 the set of all subrings of "K" containing "D". In particular, "D" is integrally closed, 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". Indeed, the integral closure is contained in the intersection since the valuation rings are integrally closed. Conversely, let "x" be in "K" but not integral over "A". Since the ideal formula_59 is not formula_60, it is contained in a maximal ideal formula_45. Then there is a valuation ring "R" that dominates the localization of formula_60 at formula_45. Since formula_64, formula_40.
The dominance is used in algebraic geometry. Let "X" be an algebraic variety over a field "k". Then we say a valuation ring "R" in formula_66 has "center "x" on "X"" if formula_38 dominates the local ring formula_68 of the structure sheaf at "x".
Ideals in valuation rings.
We may describe the ideals in the valuation ring by means of its value group.
Let Γ be a totally ordered abelian group. A subset Δ of Γ is called a "segment" if it is nonempty and, for any α in Δ, any element between -α and α is also in Δ (end points included). A subgroup of Γ is called an "isolated subgroup" if it is a segment and is a proper subgroup.
Let "D" be a valuation ring with valuation "v" and value group Γ. For any subset "A" of "D", we let formula_69 be the complement of the union of formula_70 and formula_71 in formula_72. If "I" is a proper ideal, then formula_73 is a segment of formula_72. In fact, the mapping formula_75 defines an inclusion-reversing bijection between the set of proper ideals of "D" and the set of segments of formula_72. Under this correspondence, the nonzero prime ideals of "D" correspond bijectively to the isolated subgroups of Γ.
Example: The ring of "p"-adic integers formula_17 is a valuation ring with value group formula_15. The zero subgroup of formula_15 corresponds to the unique maximal ideal formula_80 and the whole group to the zero ideal. The maximal ideal is the only isolated subgroup of formula_15.
The set of isolated subgroups is totally ordered by inclusion. The height or rank "r"(Γ) of Γ is defined to be the cardinality of the set of isolated subgroups of Γ. 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 ℝ under addition (or equivalently, of the positive real numbers ℝ+ under multiplication.) A valuation ring with a valuation of height one has a corresponding absolute value defining an ultrametric place. 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,
Places.
General definition.
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 formula_83, formula_84. The image of a place is a field called the residue field of "p". For example, the canonical map formula_85 is a place.
Example.
Let "A" be a Dedekind domain and formula_45 a prime ideal. Then the canonical map formula_87 is a place.
Specialization of places.
We say a place "p" "specializes to" a place "p", denoted by formula_88, if the valuation ring of "p" contains the valuation ring of "p". In algebraic geometry, we say a prime ideal formula_45 specializes to formula_90 if formula_91. The two notions coincide: formula_88 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 formula_93 are valuation rings of the same field, then "D" corresponds to a prime ideal of formula_94.)
Example.
For example, in the function field formula_6 of some algebraic variety formula_7 every prime ideal formula_97 contained in a maximal ideal formula_98 gives a specialization formula_99.
Remarks.
It can be shown: if formula_88, then formula_101 for some place "q" of the residue field formula_102 of "p". (Observe formula_103 is a valuation ring of formula_102 and let "q" be the corresponding place; the rest is mechanical.) 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 formula_106 is called the "center" of "p" in "A".
Places at infinity.
For the function field on an affine variety formula_7 there are valuations which are not associated to any of the primes of formula_7. These valuations are called the places at infinity. For example, the affine line formula_109 has function field formula_110. The place associated to the localization of formula_111at the maximal idealformula_112is a place at infinity.