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Given a fieldF, 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.
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 fractionsK, the following are equivalent:
For every nonzero x in K, either x in D or x-1 in D.
The equivalence of the first three definitions follows easily. A theorem of (Krull 1939) states that any ring satisfying the first three conditions satisfies the fourth: take ? to be the quotient K×/D× of the unit group of K by the unit group of D, and take ? to be the natural projection. We can turn ? into a totally ordered group by declaring the residue classes of elements of D as "positive".
Even further, given any totally ordered abelian group ?, there is a valuation ring D with value group ? (see a section below).
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".
Any field is a valuation ring. For example, the ring of rational functions on an algebraic variety .
A simple non-example is the integral domain since the inverse of a generic is
The field of power series:
has the valuation . The subring is a valuation ring as well.
the localization of the integers at the prime ideal (p), consisting of ratios where the numerator is any integer and the denominator is not divisible by p. The field of fractions is the field of rational numbers
Any ring of p-adic integers for a given prime p is a local ring, with field of fractions the p-adic numbers . The integral closure of the p-adic integers is also a local ring, with field of fractions (the algebraic closure of p-adic numbers). Both and are valuation rings.
Let k be an ordered field. An element of k is called finite if it lies between two integers n < x < m; otherwise it is called infinite. The set D of finite elements of k is a valuation ring. The set of elements x such that x ? D and x-1?D is the set of infinitesimal elements; and an element x such that x ? D and x-1 ? D is called infinite.
The ring F of finite elements of a hyperreal field *R (an ordered field containing the real numbers) is a valuation ring of *R. F consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, which is equivalent to saying a hyperreal number x such that -n < x < n for some standard integer n. The residue field, finite hyperreal numbers modulo the ideal of infinitesimal hyperreal numbers, is isomorphic to the real numbers.
For example, consider the inclusion . These are all subrings in the field of bounded-below power series .
For a given totally ordered abelian group ? and a residue fieldk, define K = k((?)) to be the ring of formal power series whose powers come from ?, that is, the elements of K are functions from ? to k such that the support (the elements of ? where the function value is not the zero of k) of each function is a well-ordered subset of ?. Addition is pointwise, and multiplication is the Cauchy product or convolution, that is the natural operation when viewing the functions as power series:
The valuation ?(f) for f in K is defined to be the least element of the support of f, that is the least element g of ? such that f(g) is nonzero. The f with ?(f)>=0 (along with 0 in K), form a subring D of K that is a valuation ring with value group ?, valuation ?, and residue field k. This construction is detailed in (Fuchs & Salce 2001, pp. 66-67), and follows a construction of (Krull 1939) which uses quotients of polynomials instead of power series.
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 ringD/M is a field, called the residue field of D.
In general, we say a local ring dominates a local ring if and ; in other words, the inclusion is a local ring homomorphism. Every 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 is nonempty and is inductive; thus, has a maximal element by Zorn's lemma. We claim R is a valuation ring. R is a local ring with maximal ideal containing by maximality. Again by maximality it is also integrally closed. Now, if , then, by maximality, and thus we can write:
Since is a unit element, this implies that is integral over R; thus is in R. This proves R is a valuation ring. (R dominates A since its maximal ideal contains 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 a ring homomorphism into an algebraically closed fieldk. Then f extends to a ring homomorphism , D some valuation ring of K containing A. (Proof: Let 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, contains a polynomial ring to which g extends, a contradiction to maximality. It follows is an algebraic field extension of . Thus, 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, . Then since dominates , which is a valuation ring since the ideals are totally ordered. This observation is subsumed to the following: there is a bijective correspondence the set of all subrings of K containing D. In particular, D is integrally closed, and the Krull dimension of D is the cardinality 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 is not , it is contained in a maximal ideal . Then there is a valuation ring R that dominates the localization of at . Since , .
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 has "center x on X" if dominates the local ring 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 ? be a totally ordered abelian group. A subset ? of ? is called a segment if it is nonempty and, for any ? in ?, any element between -? and ? is also in ? (end points included). A subgroup of ? 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 ?. For any subset A of D, we let be the complement of the union of and in . If I is a proper ideal, then is a segment of . In fact, the mapping defines an inclusion-reversing bijection between the set of proper ideals of D and the set of segments of . Under this correspondence, the nonzero prime ideals of D correspond bijectively to the isolated subgroups of ?.
Example: The ring of p-adic integers is a valuation ring with value group . The zero subgroup of corresponds to the unique maximal ideal and the whole group to the zero ideal. The maximal ideal is the only isolated subgroup of .
The set of isolated subgroups is totally ordered by inclusion. The height or rankr(?) of ? is defined to be the cardinality of the set of isolated subgroups of ?. Since the nonzero prime ideals are totally ordered and they correspond to isolated subgroups of ?, the height of ? is equal to the Krull dimension of the valuation ring D associated with ?.
The rational rankrr(?) is defined as the rank of the value group as an abelian group,
The reference to this section is Zariski-Samuel.
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 , . The image of a place is a field called the residue field of p. For example, the canonical map is a place.
Let A be a Dedekind domain and a prime ideal. Then the canonical map is a place.
Specialization of places
We say a place pspecializes to a place p', denoted by , if the valuation ring of p contains the valuation ring of p'. In algebraic geometry, we say a prime ideal specializes to if . The two notions coincide: 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 .)
For example, in the function field of some algebraic variety every prime ideal contained in a maximal ideal gives a specialization .
It can be shown: if , then for some place q of the residue field of p. (Observe is a valuation ring of 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 is called the center of p in A.
Places at infinity
For the function field on an affine variety there are valuations which are not associated to any of the primes of . These valuations are called the places at infinity. For example, the affine line has function field . The place associated to the localization of
^Proof: if R is a maximal element, then it is dominated by a valuation ring; thus, it itself must be a valuation ring. Conversely, let R be a valuation ring and S a local ring that dominates R but not R. There is x that is in S but not in R. Then is in R and in fact in the maximal ideal of R. But then , which is absurd. Hence, there cannot be such S.
^To see more directly that valuation rings are integrally closed, suppose that xn + a1xn − 1 + ... + a0 = 0. Then dividing by
xn−1 gives us x = − a1 − ... − a0x − n + 1. If x were not in D, then x -1 would be in D and this would express x as a finite sum of elements in D, so that x would be in D, a contradiction.