Valuation ring
Encyclopedia
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 x in F, then D is said to be a valuation ring for the field F or a place of F. Since F is in this case 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 ideal
s are totally ordered by inclusion; or equivalently their principal ideal
s are totally ordered by inclusion. The valuation rings of a field are the maximal elements of the local subrings partially ordered by dominance, where
dominates 
if 
and 
.
In particular, every valuation ring is a local ring
.
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 G 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. Even further, given any totally ordered abelian group G, there is a valuation ring D with value group G.
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
".
s, 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.
under multiplication, which is a subgroup
of the units
F* of F, the nonzero elements of F. These are both abelian group
s, and we can define the quotient group
V = F*/D*, which is the value group of D. Hence, we have a group homomorphism
ν from F* to the value group V. It is customary to write the group operation in V as +.
We can turn V into a totally ordered group
by declaring the residue classes of elements of D as "positive". More precisely, V is totally ordered by defining
if and only if
where [x] and [y] are equivalence classes in V.
These are precisely the properties of a valuation, and the study of valuations is essentially the study of valuation rings.
k, define K = k((G)) to be the ring of formal power series whose powers come from G, that is, the elements of K are functions from G to k such that the support
(the elements of G 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:
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 G 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 G, valuation ν, and residue field k. This construction is detailed in , and follows a construction of which uses quotients of polynomials instead of power series.
defining an ultrametric place. There is a correspondence of the isolated proper subgroups of the value group of a valuation and the prime ideals of the valuation ring. This implies that a valuation ring is discrete if and only if it is Noetherian.
The rational rank rr(V) is defined as the rank of the value group as an abelian group
.
Here, an integral domain D which is integrally closed in its field of fractions is said to be integrally closed. This means that if a member x of the field of fractions F of D satisfies an equation of the form xn + a1xn−1 + ... + a0 = 0, where the coefficients ai are elements of D, then x is in D.
To see 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.
, or PID, is an integral domain in which every ideal is a principal ideal. A PID with only one non-zero maximal ideal
is called a discrete valuation ring
, or DVR, and every discrete valuation ring is a valuation ring. A valuation ring is a PID if and only if it is a DVR or a field. A value group is called discrete if and only 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.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a valuation ring is an integral domain D such that for every element x of its field of fractions
Field of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
F, at least one of x or x −1 belongs to D.
Given a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
F, if D is a subring
Subring
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
of F such that either x or x −1 belongs to
D for every x in F, then D is said to be a valuation ring for the field F or a place of F. Since F is in this case 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 ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
s are totally ordered by inclusion; or equivalently their principal ideal
Principal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
s are totally ordered by inclusion. The valuation rings of a field are the maximal elements of the local subrings partially ordered by dominance, where
dominates if and .In particular, every valuation ring is a local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
.
Examples
- Any field is a valuation ring.
- Z(p), the localizationLocalization of a ringIn abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
of the integers Z 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 Q.
- The ring of meromorphic functionMeromorphic functionIn complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s on the entire complex planeComplex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
which have a Maclaurin series (Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
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 Zp for a given prime p is a local ringLocal ringIn abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
, with field of fractions the p-adic numbers Qp. The algebraic closureAlgebraic closureIn mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
Zpcl of the p-adic integers is also a local ring, with field of fractions Qpcl. Both Zp and Zpcl are valuation rings.
- Let k be an ordered fieldOrdered fieldIn mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
. An element of k is called finite if it lies between two integers n
- The ring F of finite elements of a hyperreal fieldHyperreal numberThe system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
*R 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 fieldResidue fieldIn mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field...
, finite hyperreal numbers modulo the ideal of infinitesimal hyperreal numbers, is isomorphic to the real numbers.
Definitions
There are several equivalent definitions of valuation ring. For a subring D of its field of fractionsField of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
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
- There is a totally ordered abelian groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
G (called the value group) and a surjective group homomorphism (called the valuation) ν:K×→G with D = { x in K : ν(x) ≥ 0 } ∪ {0}
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 G 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. Even further, given any totally ordered abelian group G, there is a valuation ring D with value group G.
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
Serial module
In abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either N_1\subseteq N_2 or N_2\subseteq N_1. A module is called a serial module if it is a direct sum of uniserial...
".
Units and maximal ideals
The unitUnit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
s, 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
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
, the quotient ring
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
D/M is a field, called the residue field of D.
Value group
The units D* of D comprise a groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
under multiplication, which is a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of the units
F* of F, the nonzero elements of F. These are both abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s, and we can define the quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
V = F*/D*, which is the value group of D. Hence, we have a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
ν from F* to the value group V. It is customary to write the group operation in V as +.
We can turn V into a totally ordered group
Ordered group
In abstract algebra, a partially-ordered group is a group equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b.An element x of G is called positive element if 0 ≤ x...
by declaring the residue classes of elements of D as "positive". More precisely, V is totally ordered by defining
if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
where [x] and [y] are equivalence classes in V.Valuation
We add to V the special value ∞, defined to be larger than any other element of V, and such that x+∞ = ∞ for all x. If we then define ν(0) = ∞, i.e., making zero larger in value than anything else, we have the following properties:- ν(x) = ∞ if and only if x=0
- ν(xy) = ν(x) + ν(y)
- ν(x+y) ≥ min(ν(x), ν(y))
These are precisely the properties of a valuation, and the study of valuations is essentially the study of valuation rings.
Construction
For a given totally ordered abelian group G and a residue fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
k, define K = k((G)) to be the ring of formal power series whose powers come from G, that is, the elements of K are functions from G to k such that the support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
(the elements of G 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:
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 G 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 G, valuation ν, and residue field k. This construction is detailed in , and follows a construction of which uses quotients of polynomials instead of power series.
Height of a value group
If V is a totally ordered group, a subgroup U of G is called an isolated subgroup of G if 0 ≤ y ≤ x and x ∈ U implies y ∈ U. The set of isolated subgroups is totally ordered by inclusion. The height or rank r(V) of V is defined to be the cardinality of the set of proper isolated subgroups of V. The most important special case is height one, which is equivalent to V being a subgroup of the real numbers under addition (or equivalently, of the positive real numbers under multiplication.) A value ring with a valuation of height one has a corresponding absolute valueAbsolute value (algebra)
In mathematics, an absolute value is a function which measures the "size" of elements in a field or integral domain. More precisely, if D is an integral domain, then an absolute value is any mapping | x | from D to the real numbers R satisfying:* | x | ≥ 0,*...
defining an ultrametric place. There is a correspondence of the isolated proper subgroups of the value group of a valuation and the prime ideals of the valuation ring. This implies that a valuation ring is discrete if and only if it is Noetherian.
The rational rank rr(V) is defined as the rank of the value group as an abelian group
.Integral closure
A valuation ring is always integrally closed.Here, an integral domain D which is integrally closed in its field of fractions is said to be integrally closed. This means that if a member x of the field of fractions F of D satisfies an equation of the form xn + a1xn−1 + ... + a0 = 0, where the coefficients ai are elements of D, then x is in D.
To see 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.
Principal ideal domains
A principal ideal domainPrincipal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
, or PID, is an integral domain in which every ideal is a principal ideal. A PID with only one non-zero maximal ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
is called a discrete valuation ring
Discrete valuation ring
In abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...
, or DVR, and every discrete valuation ring is a valuation ring. A valuation ring is a PID if and only if it is a DVR or a field. A value group is called discrete if and only 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.