Discrete valuation ring

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

This means a DVR is an integral domain R that satisfies any one of the following equivalent conditions:


 * 1) R is a local principal ideal domain, and not a field.
 * 2) R is a valuation ring with a value group isomorphic to the integers under addition.
 * 3) R is a local Dedekind domain and not a field.
 * 4) R is a Noetherian local domain whose maximal ideal is principal, and not a field.
 * 5) R is an integrally closed Noetherian local ring with Krull dimension one.
 * 6) R is a principal ideal domain with a unique non-zero prime ideal.
 * 7) R is a principal ideal domain with a unique irreducible element (up to multiplication by units).
 * 8) R is a unique factorization domain with a unique irreducible element (up to multiplication by units).
 * 9) 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.
 * 10) 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}.

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 $2^{k} z⁄n$ 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
 * $$\mathbb Z_{(p)}:=\left.\left\{\frac zn\,\right| z,n\in\mathbb Z,p\nmid n\right\}$$

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 $$R = kT$$ 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.

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 (tk) 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 αtk 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.

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

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 $$\Z_{(p)}=\Q \cap \Z_p$$ (which can be seen as the set of all rational numbers that are p-adic integers) is the ring of all p-adic integers Zp.