Tags: #todo #todo/stub Refs: ?
At a prime ideal:
Localization for modules as base change: 
When is \(M\) a submodule of one of its localizations? 
# Examples
Valuations for Laurent series 
Localizations in \(k[x]\), and function evaluation as a morphism onto the residue field: 
The most interesting commutative algebras are \(S/I\) where \(S = k[x_1, \cdots, x_n]\) for \(k\) a field, \({\mathbb{Z}}\), or the localization of a ring at a prime ideal.
For a nonarchimedean field with an absolute value \({\left\lvert {{-}} \right\rvert}\), the ring of integers are given by the valuation ring, i.e. the closed disc: \begin{align*} {\mathcal{O}}_K \coloneqq\left\{{ x\in K {~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert} \leq 1}\right\} = \left\{{x\in K {~\mathrel{\Big\vert}~}v(x) \geq 0}\right\} = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu_K \subseteq K \end{align*} Its units are given by the boundary sphere: \begin{align*} {\mathcal{O}}_K^{\times}\coloneqq\left\{{x\in K {~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert} = 1}\right\} = \left\{{x\in K {~\mathrel{\Big\vert}~}v(x) = 0}\right\} = {{\partial}}\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu_K \end{align*} This is a local ring with maximal ideal the open disc: \begin{align*} {\mathfrak{m}}\coloneqq\left\{{x\in K {~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert} < 1 }\right\}= {\mathbb{D}}_K \end{align*}
What is the localization of a ring?
