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,[1] where

dominates if and .[2]

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.

DefinitionsEdit

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:

  1. For every nonzero x in K, either x in D or x−1 in D.
  2. The ideals of D are totally ordered by inclusion.
  3. The principal ideals of D are totally ordered by inclusion (i.e., the elements in D are totally ordered by divisibility.)
  4. 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 = { xK× | 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".[3]

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.[4] 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.[5]

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".

ExamplesEdit

  • Any field   is a valuation ring.
  • 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  
  • The ring of meromorphic functions on the entire complex plane which have a Maclaurin series (Taylor series expansion at zero) is a valuation ring. The field of fractions are the functions meromorphic on the whole plane. If f does not have a Maclaurin series then 1/f does.
  • 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 xD and x−1D is the set of infinitesimal elements; and an element x such that xD and x−1D 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.
  • A common geometric example comes from algebraic plane curves. Consider the polynomial ring   and an irreducible polynomial   in that ring. Then the ring   is the ring of polynomial functions on the curve  . Choose a point   such that   and it is a regular point on the curve; i.e., the local ring R at the point is a regular local ring of Krull dimension one or a discrete valuation ring.
  • For example, consider the inclusion  . These are all subrings in the field of bounded-below power series  .

ConstructionEdit

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 Γ. Addition is pointwise, and multiplication is the Cauchy product or convolution, that is the natural operation when viewing the functions as power series:

  with  

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 closureEdit

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   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.[6]

Let A be a subring of a field K and   a ring homomorphism into an algebraically closed field k. 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:[7] there is a bijective correspondence   the set of all subrings of K containing D. In particular, D is integrally closed,[8][9] 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.[10] 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  ,[11] 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.[12]

Ideals in valuation ringsEdit

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   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  .[13] Under this correspondence, the nonzero prime ideals of D correspond bijectively to the isolated subgroups of Γ.

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 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+ 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,

 

PlacesEdit

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.

Example: Let A be a Dedekind domain and   a prime ideal. Then the canonical map   is a place.

We say a place p specializes 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  .)

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.

NotesEdit

  1. ^ Hartshorne 1977, Theorem I.6.1A
  2. ^ Efrat (2006) p.55
  3. ^ More precisely, Γ is totally ordered by defining   if and only if   where [x] and [y] are equivalence classes in Γ. cf. Efrat (2006) p.39
  4. ^ Cohn 1968, Proposition 1.5
  5. ^ Efrat (2006) p.43
  6. ^ 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.
  7. ^ Zariski−Samuel, Ch. VI, Theorem 3
  8. ^ Efrat (2006) p.38
  9. ^ 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.
  10. ^ Matsumura 1986, Theorem 10.4
  11. ^ In general,   is integral over A if and only if  
  12. ^ Hartshorne 1977, Ch II. Exercise 4.5
  13. ^ Zariski−Samuel, Ch. VI, Theorem 15

ReferencesEdit

  • Nicolas Bourbaki, Commutative Algebra, Addison-Wesley, 1972
  • Cohn, P. M. (1968), "Bezout rings and their subrings" (PDF), Proc. Cambridge Philos. Soc., 64: 251–264, doi:10.1017/s0305004100042791, ISSN 0008-1981, MR 0222065, Zbl 0157.08401
  • Efrat, Ido (2006), Valuations, orderings, and Milnor K-theory, Mathematical Surveys and Monographs, 124, Providence, RI: American Mathematical Society, ISBN 0-8218-4041-X, Zbl 1103.12002
  • Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR 1794715, Zbl 0973.13001
  • Krull, Wolfgang (1939), "Beiträge zur Arithmetik kommutativer Integritätsbereiche. VI. Der allgemeine Diskriminantensatz. Unverzweigte Ringerweiterungen", Mathematische Zeitschrift, 45 (1): 1–19, doi:10.1007/BF01580269, ISSN 0025-5874, MR 1545800, Zbl 0020.34003
  • Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
  • Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Translated from the Japanese by Miles Reid (Second ed.), ISBN 0-521-36764-6, Zbl 0666.13002
  • Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876