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.
Contents
- Examples
- Definitions
- Construction
- Dominance and integral closure
- Ideals in valuation rings
- Places
- References
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.
Examples
Definitions
There are several equivalent definitions of valuation ring (see below for the characterization in terms of dominance). For a subring D of its field of fractions K the following are equivalent:
- For every nonzero x in K, either x in D or x−1 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 totally ordered by divisibility.)
- There is a totally ordered abelian group Γ (called the value group) and a surjective group homomorphism (called the valuation) ν:K× → Γ with D = { x in K× : ν(x) ≥ 0 } ∪ {0}
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 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 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".
Construction
For a given totally ordered abelian group Γ and a residue field k, define K = k((Γ)) to be the ring of formal power series whose powers come from Γ, that is, the elements of K are functions from Γ to k such that the support (the elements of Γ where the function value is not the zero of k) of each function is a well-ordered subset of G. 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 ν(f) for f in K is defined to be the least element of the support of f, that is the least element g of Γ such that f(g) is nonzero. The f with ν(f)≥0 (along with 0 in K), form a subring D of K that is a valuation ring with value group Γ, valuation ν, 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 ring D/M is a field, called the residue field of D.
In general, we say a local ring
Since
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
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,
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
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
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
Example: The ring of p-adic integers
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
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
Example: Let A be a Dedekind domain and
We say a place p specializes to a place p', denoted by
It can be shown: if
If p is a place and A is a subring of the valuation ring of p, then