Discrete valuation ring

Last updated November 07, 2024

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

R is a local principal ideal domain, and not a field.
R is a valuation ring with a value group isomorphic to the integers under addition.
R is a local Dedekind domain and not a field.
R is a Noetherian local domain whose maximal ideal is principal, and not a field.^[1]
R is an integrally closed Noetherian local ring with Krull dimension one.
R is a principal ideal domain with a unique non-zero prime ideal.
R is a principal ideal domain with a unique irreducible element (up to multiplication by units).
R is a unique factorization domain with a unique irreducible element (up to multiplication by units).
R is Noetherian, not a field, and every nonzero fractional ideal of R is irreducible in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it.
There is some discrete valuation ν on the field of fractions K of R such that R = {0} $\cup$ {x $\in$ K : ν(x) ≥ 0}.

Examples

Algebraic

Localization of Dedekind rings

Let $\mathbb {Z} _{(2)}:=\{z/n\mid z,n\in \mathbb {Z} ,\,\,n{\text{ is odd}}\}$ . Then, the field of fractions of $\mathbb {Z} _{(2)}$ is $\mathbb {Q}$ . For any nonzero element $r$ of $\mathbb {Q}$ , we can apply unique factorization to the numerator and denominator of r to write r as ⁠2^kz/n⁠ where z, n, and k are integers with z and n odd. In this case, we define ν(r)=k. Then $\mathbb {Z} _{(2)}$ is the discrete valuation ring corresponding to ν. The maximal ideal of $\mathbb {Z} _{(2)}$ is the principal ideal generated by 2, i.e. $2\mathbb {Z} _{(2)}$ , and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter). Note that $\mathbb {Z} _{(2)}$ is the localization of the Dedekind domain $\mathbb {Z}$ at the prime ideal generated by 2.

More generally, any localization of a Dedekind domain at a non-zero prime ideal is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise. In particular, we can define rings

\mathbb {Z} _{(p)}:=\left.\left\{{\frac {z}{n}}\,\right|z,n\in \mathbb {Z} ,p\nmid n\right\}

for any prime p in complete analogy.

p-adic integers

The ring $\mathbb {Z} _{p}$ of p-adic integers is a DVR, for any prime $p$ . Here $p$ is an irreducible element; the valuation assigns to each $p$ -adic integer $x$ the largest integer $k$ such that $p^{k}$ divides $x$ .

Formal power series

Another important example of a DVR is the ring of formal power series $R=k[[T]]$ in one variable $T$ over some field $k$ . The "unique" irreducible element is $T$ , the maximal ideal of $R$ is the principal ideal generated by $T$ , and the valuation $\nu$ assigns to each power series the index (i.e. degree) of the first non-zero coefficient.

If we restrict ourselves to real or complex coefficients, we can consider the ring of power series in one variable that converge in a neighborhood of 0 (with the neighborhood depending on the power series). This is a discrete valuation ring. This is useful for building intuition with the Valuative criterion of properness.

Ring in function field

For an example more geometrical in nature, take the ring R = {f/g : f, g polynomials in R[X] and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the "unique" irreducible element is X and the valuation assigns to each function f the order (possibly 0) of the zero of f at 0. This example provides the template for studying general algebraic curves near non-singular points, the algebraic curve in this case being the real line.

Scheme-theoretic

Henselian trait

For a DVR $R$ it is common to write the fraction field as $K={\text{Frac}}(R)$ and $\kappa =R/{\mathfrak {m}}$ the residue field. These correspond to the generic and closed points of $S={\text{Spec}}(R).$ For example, the closed point of ${\text{Spec}}(\mathbb {Z} _{p})$ is $\mathbb {F} _{p}$ and the generic point is $\mathbb {Q} _{p}$ . Sometimes this is denoted as

\eta \to S\leftarrow s

where $\eta$ is the generic point and $s$ is the closed point .

Localization of a point on a curve

Given an algebraic curve $(X,{\mathcal {O}}_{X})$ , the local ring ${\mathcal {O}}_{X,{\mathfrak {p}}}$ at a smooth point ${\mathfrak {p}}$ is a discrete valuation ring, because it is a principal valuation ring. Note because the point ${\mathfrak {p}}$ is smooth, the completion of the local ring is isomorphic to the completion of the localization of $\mathbb {A} ^{1}$ at some point ${\mathfrak {q}}$ .

Uniformizing parameter

Given a DVR R, any irreducible element of R is a generator for the unique maximal ideal of R and vice versa. Such an element is also called a uniformizing parameter of R (or a uniformizing element, a uniformizer, or a prime element).

If we fix a uniformizing parameter t, then M=(t) is the unique maximal ideal of R, and every other non-zero ideal is a power of M, i.e. has the form (t^k) for some k≥0. All the powers of t are distinct, and so are the powers of M. Every non-zero element x of R can be written in the form αt^k with α a unit in R and k≥0, both uniquely determined by x. The valuation is given by ν(x) = kv(t). So to understand the ring completely, one needs to know the group of units of R and how the units interact additively with the powers of t.

The function v also makes any discrete valuation ring into a Euclidean domain.^{[ citation needed ]}

Topology

Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ring. We can also give it a metric space structure where the distance between two elements x and y can be measured as follows:

|x-y|=2^{-\nu (x-y)}

(or with any other fixed real number > 1 in place of 2). Intuitively: an element z is "small" and "close to 0" iff its valuation ν(z) is large. The function |x-y|, supplemented by |0|=0, is the restriction of an absolute value defined on the field of fractions of the discrete valuation ring.

A DVR is compact if and only if it is complete and its residue field R/M is a finite field.

Examples of complete DVRs include

the ring of p-adic integers and
the ring of formal power series over any field

For a given DVR, one often passes to its completion, a complete DVR containing the given ring that is often easier to study. This completion procedure can be thought of in a geometrical way as passing from rational functions to power series, or from rational numbers to the reals.

Returning to our examples: the ring of all formal power series in one variable with real coefficients is the completion of the ring of rational functions defined (i.e. finite) in a neighborhood of 0 on the real line; it is also the completion of the ring of all real power series that converge near 0. The completion of $\mathbb {Z} _{(p)}=\mathbb {Q} \cap \mathbb {Z} _{p}$ (which can be seen as the set of all rational numbers that are p-adic integers) is the ring of all p-adic integers Z_p.

Related Research Articles

In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal. Some authors such as Bourbaki refer to PIDs as principal rings.

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In number theory, given a prime number $p$ , the $p$ -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; $p$ -adic numbers can be written in a form similar to decimals, but with digits based on a prime number $p$ rather than ten, and extending to the left rather than to the right.

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

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In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.

In mathematics, a field K is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation v and if its residue field k is finite. In general, a local field is a locally compact topological field with respect to a non-discrete topology. The real numbers R, and the complex numbers C are Archimedean local fields. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields.

In mathematics, more specifically 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 algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.

<span class="mw-page-title-main">Algebraic number theory</span> Branch of number theory

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In algebra, a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.

In mathematics, the ring of integers of an algebraic number field $is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of, the ring is always a subring of .$

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.

In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function:

In abstract algebra, a valuation ring is an integral domain D such that for every non-zero element x of its field of fractions F, at least one of x or x⁻¹ belongs to D.

In algebra, 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:

In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions R on a space X concentrates on a formal neighborhood of a point of X: heuristically, this is a neighborhood so small that all Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when R has a metric given by a non-Archimedean absolute value.

In commutative algebra, an integrally closed domainA is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A that is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied domains are integrally closed, as shown by the following chain of class inclusions:

In mathematics, an algebraic number field is an extension field $of the field of rational numbers such that the field extension has finite degree . Thus is a field that contains and has finite dimension when considered as a vector space over .$

References

↑ "ac.commutative algebra - Condition for a local ring whose maximal ideal is principal to be Noetherian". MathOverflow.

Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: John Wiley & Sons, ISBN 978-0-471-43334-7, MR 2286236
Discrete valuation ring, The Encyclopaedia of Mathematics .

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[1] "ac.commutative algebra - Condition for a local ring whose maximal ideal is principal to be Noetherian". MathOverflow.

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