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 "x" in "F", then "D" is said to be a valuation ring for the field "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 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.

Examples

* Any field is a valuation ring.

* Z_("p"), the localization 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 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 Z_{"p" for a given prime "p" is a local ring, with field of fractions the "p"-adic numbers Qp. The algebraic closure Z"p"^cl of the "p"-adic integers is also a local ring, with field of fractions Q"p"^cl. Both Z"p" and Z"p"^cl 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 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.

Units and maximal ideals

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".}

Value group

The units "D"^* of "D" comprise a group under multiplication, which is a subgroup of the units "F"^* of "F", the nonzero elements of "F". These are both abelian groups, 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 $[x]\; geq\; [y]$ if and only if $x\; y^\{\; -\; 1\; \}\; in\; D$ 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) = ∞, making zero larger in value than anything else, we have the following properties:

* ν("x") ≤ ∞
* 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.

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 height of "V" is defined to be one less than the cardinality of the set of isolated subgroups of "V". If "V" is the trivial group, corresponding to the trivial valuation resulting from taking a field as a valuation ring, then "V" is of height zero. 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 value defining an ultrametric place.

Integral closure

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 "x"^"n"+"a"₁"x"^"n"-1+...+"a"₀ = 0, where the coefficients "a"_"i" are elements of "D", then "x" is in "D". A valuation ring is always integrally closed; if "x"^"n"+"a"₁"x"^"n"-1+...+"a"₀ = 0 then dividing by"x"^"n"-1 gives us "x" = -"a"₁-...-"a"₀"x"^-"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 domain, 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.

References

* Nicolas Bourbaki, "Commutative Algebra", Addison-Wesley, 1972