Discrete valuation ring

This means a DVR is an integral domain ${\displaystyle R}$ that satisfies any and all of the following equivalent conditions:

1. ${\displaystyle R}$ is a local ring, a principal ideal domain, and not a field.
2. ${\displaystyle R}$ is a valuation ring with a value group isomorphic to the integers under addition.
3. ${\displaystyle R}$ is a local ring, a Dedekind domain, and not a field.
4. ${\displaystyle R}$ is Noetherian and a local domain whose unique maximal ideal is principal, and not a field.
5. ${\displaystyle R}$ is integrally closed, Noetherian, and a local ring with Krull dimension one.
6. ${\displaystyle R}$ is a principal ideal domain with a unique non-zero prime ideal.
7. ${\displaystyle R}$ is a principal ideal domain with a unique irreducible element (up to multiplication by units).
8. ${\displaystyle R}$ is a unique factorization domain with a unique irreducible element (up to multiplication by units).
9. ${\displaystyle R}$ 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 ${\displaystyle \nu }$ on the field of fractions ${\displaystyle K}$ of ${\displaystyle R}$ such that ${\displaystyle R=\{0\}\cup \{x\in K:\nu (x)\ge 0\}}$.
11. ${\displaystyle R}$ is a Noetherian local ring whose unique maximal ideal is generated by a single non-nilpotent element.

Examples

Algebraic

Localization of Dedekind rings

Let ${\displaystyle {\mathbb{Z}}_{(2)}}$ be the localization of ${\displaystyle \mathbb{Z}}$ at the ideal generated by 2. Formally,

${\displaystyle {\mathbb{Z}}_{(2)}=\{z/n:z,n\in \mathbb{Z},n\text{ is odd}\}.}$

The field of fractions of ${\displaystyle {\mathbb{Z}}_{(2)}}$ is ${\displaystyle \mathbb{Q}}$. For any nonzero element ${\displaystyle r}$ of ${\displaystyle \mathbb{Q}}$, we can apply unique factorization to the numerator and denominator of ${\displaystyle r}$ to write ${\displaystyle r}$ as ${\displaystyle \frac{2^{k}z}{n}}$ where ${\displaystyle z}$, ${\displaystyle n}$, and ${\displaystyle k}$ are integers with ${\displaystyle z}$ and ${\displaystyle n}$ odd. In this case, we define ${\displaystyle \nu (r)=k}$.

Then ${\displaystyle {\mathbb{Z}}_{(2)}}$ is the discrete valuation ring corresponding to ${\displaystyle \nu }$. The maximal ideal of ${\displaystyle {\mathbb{Z}}_{(2)}}$ is the principal ideal generated by 2; i.e., ${\displaystyle 2{\mathbb{Z}}_{(2)}}$, and the "unique" irreducible element (up to units) is 2 (also known as a uniformizing parameter).

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

${\displaystyle {\mathbb{Z}}_{(p)}:=\{z/n:z,n\in \mathbb{Z},p\nmid n\}}$

for any prime ${\displaystyle p}$ in complete analogy.

p-adic integers

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

Formal power series

Another important example of a DVR is the ring of formal power series ${\displaystyle R=k[[T]]}$ in one variable ${\displaystyle T}$ over some field ${\displaystyle k}$. The "unique" irreducible element is ${\displaystyle T}$, the maximal ideal of ${\displaystyle R}$ is the principal ideal generated by ${\displaystyle T}$, and the valuation ${\displaystyle \nu }$ 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

${\displaystyle R=\{f/g:f,g\in \mathbb{R}[x],g(0)\ne 0\},}$

considered as a subring of the field of rational functions ${\displaystyle \mathbb{R}(x)}$. ${\displaystyle R}$ can be identified with the ring of all real-valued rational functions defined (i.e., finite) on 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 ${\displaystyle x}$ and the valuation assigns to each function ${\displaystyle f}$ the order (possibly 0) of the zero of ${\displaystyle f}$ 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 ${\displaystyle R}$ it is common to write the fraction field as ${\displaystyle K=\text{Frac}(R)}$ and ${\displaystyle \kappa =R/\mathfrak{m}}$ the residue field. These correspond to the generic and closed points of ${\displaystyle S=\text{Spec}(R).}$ For example, the closed point of ${\displaystyle \text{Spec}({\mathbb{Z}}_p)}$ is ${\displaystyle {\mathbb{F}}_p}$ and the generic point is ${\displaystyle {\mathbb{Q}}_p}$. Sometimes this is denoted as

${\displaystyle \eta \to S\leftarrow s}$

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

Localization of a point

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

For any nonsingular variety, one can similarly consider the local ring at a point of codimension one. More generally, on a (locally) Noetherian normal scheme, the local ring at the generic point of a prime Weil divisor is a discrete valuation ring, with the discrete valuation giving the order of vanishing of a regular function along the divisor.