Discrete valuation ring

Let ${\displaystyle \mathbb {Z} _{(2)}:=\{z/n\mid z,n\in \mathbb {Z} ,\,\,n{\text{ is odd}}\}}$. Then, 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 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 ${\displaystyle \mathbb {Z} _{(2)}}$ is the discrete valuation ring corresponding to ν. 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 (this is also known as a uniformizing parameter). Note that ${\displaystyle \mathbb {Z} _{(2)}}$ is the localization of the Dedekind domain ${\displaystyle \mathbb {Z} }$ 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

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

for any prime p in complete analogy.

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 ${\displaystyle x}$.

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.

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.

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 .

Given an algebraic curve ${\displaystyle (X,{\mathcal {O}}_{X})}$, the local ring ${\displaystyle {\mathcal {O}}_{X,{\mathfrak {p}}}}$ at a smooth 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 ${\displaystyle {\mathfrak {q}}}$.