Editing Local ring (section)

=== Valuation theory ===
{{main|Valuation (algebra)}}

Local rings play a major role in valuation theory. By definition, a [[valuation ring]] of a field ''K'' is a subring ''R'' such that for every non-zero element ''x'' of ''K'', at least one of ''x'' and ''x''<sup>&minus;1</sup> is in ''R''. Any such subring will be a local ring. For example, the ring of [[rational number]]s with [[odd number|odd]] denominator (mentioned above) is a valuation ring in  <math>\mathbb{Q}</math>.

Given a field ''K'', which may or may not be a  [[Function field of an algebraic variety|function field]], we may look for local rings in it. If ''K'' were indeed the function field of an [[algebraic variety]] ''V'', then for each point ''P'' of ''V'' we could try to define a valuation ring ''R'' of functions "defined at" ''P''. In cases where ''V'' has dimension 2 or more there is a difficulty that is seen this way: if ''F'' and ''G'' are rational functions on ''V'' with

:''F''(''P'') = ''G''(''P'') = 0,

the function

:''F''/''G''

is an [[indeterminate form]] at ''P''. Considering a simple example, such as

:''Y''/''X'',

approached along a line

:''Y'' = ''tX'',

one sees that the ''value at'' ''P'' is a concept without a simple definition. It is replaced by using valuations.