Editing Valuation ring (section)

== Examples ==
* Any field <math>\mathbb{F}</math> is a valuation ring. For example, the field of rational functions <math>\mathbb{F}(X)</math> on an [[algebraic variety]] <math>X</math>.<ref>[https://math.stackexchange.com/q/946933 The role of valuation rings in algebraic geometry]</ref><ref>[https://mathoverflow.net/q/28595 Does there exist a Riemann surface corresponding to every field extension? Any other hypothesis needed?]</ref>
* A simple non-example is the integral domain <math>\Complex[X]</math> since the inverse of a generic <math>f/g \in \Complex(X)</math> is <math>g/f \not\in \Complex[X]</math>.
* The field of [[power series]]:
::<math>\mathbb{F}((X)) =\left\{ f(X) =\! \sum_{i>-\infty}^\infty a_iX^i \, :\ a_i \in \mathbb{F} \right\}</math>
:has the [[valuation (algebra)|valuation]] <math>v(f) = \inf\nolimits_{a_n \neq 0} n</math>. The subring <math>\mathbb{F}[[X]]</math> is a valuation ring as well.

* <math>\Z_{(p)},</math> the [[localization of a ring|localization]] of the integers <math>\Z</math> 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 <math>\Q.</math>
* The ring of [[meromorphic function]]s 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 integer|''p''-adic integers]] <math>\Z_p</math> for a given [[prime number|prime]] ''p'' is a [[local ring]], with field of fractions the [[p-adic number|''p''-adic numbers]] <math>\Q_p</math>. The [[integral closure]] <math>\Z_p^{\text{cl}}</math> of the ''p''-adic integers is also a local ring, with field of fractions <math>\Q_p^{\text{cl}}</math> (the [[algebraic closure]] of the ''p''-adic numbers). Both <math>\Z_p</math> and <math>\Z_p^{\text{cl}}</math> 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''<sup>−1</sup> ∉ ''D'' is the set of [[infinitesimal]] elements; and an element ''x'' such that ''x'' ∉ ''D'' and ''x''<sup>−1</sup> ∈ ''D'' is called infinite.
* The ring '''F''' of finite elements of a [[hyperreal number|hyperreal field]] *'''R''' (an ordered field containing the [[real number]]s) 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.
* A common geometric example comes from [[algebraic plane curve]]s.  Consider the [[polynomial ring]] <math>\Complex[x, y]</math> and an [[irreducible polynomial]] <math>f</math> in that ring. Then the ring <math>\Complex[x, y] / (f)</math> is the ring of polynomial functions on the curve <math>\{(x, y) : f(x, y) = 0\}</math>.  Choose a point <math>P = (P_x, P_y) \in \Complex ^2</math> such that <math>f(P) = 0</math> and it is a [[regular point]] on the curve; i.e., the local ring ''R'' at the point is a [[regular local ring]] of [[Krull dimension]] one or a [[discrete valuation ring]].<!-- by assumption, it is a valuation ring Then the [[localization of a ring|localization]] <math>\Complex[x, y]_{(x - P_x, y - P_y)} / (f)</math> is a discrete valuation ring. This ring represents [[germ (mathematics)|germs]] of polynomial functions at the point <math>P</math>.  The fraction field is the ring of germs of rational functions at <math>P</math>, and the valuation ring property means that if a rational function has a pole at <math>P</math>, then its reciprocal does not. The valuation of a polynomial <math>g</math> can be computed as the length of the [[Artinian ring]] <math>\Complex[x, y]_{(x-P_x, y-P_y)} / (f, g)</math> (assuming that ''g'' is in a general position) and the valuation of a quotient <math>g / h</math> can be computed as the difference of valuations.-->
* For example, consider the inclusion <math>(\mathbb{C}[[X^2]],(X^2)) \hookrightarrow (\mathbb{C}[[X]],(X))</math>. These are all subrings in the field of bounded-below power series <math>\mathbb{C}((X))</math>.