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==Definition==

===Definition with symbols===

Let <math>A</math> be a [[commutative unital ring]] and <math>P</math> a [[prime ideal]] in <math>A</math>. Then, the localization of <math>A</math> at <math>P</math> is defined as follows:

* As a set, it is the collection of fractions <math>a/s</math> where <math>a \in A, s \in A \setminus P</math>, subject to the equivalence <math>a/s \sim a'/s' \iff as' = a's</math>.
* The operations are defined as follows: <math>a/s + a'/s' = (as' + a's)/(ss')</math> and <math>(a/s) (a'/s') = (aa')/(ss')</math>

==Facts==

<math>A</math> embeds naturally as a subset of <math>A_P</math>. If <math>P \le P'</math> are prime ideals in <math>A</math>, we have an embedding from <math>A_{P'}</math> into <math>A_P</math>. In fact, if <math>A</math> is an [[integral domain], then all the <math>A_P</math>s are contained inside the residue field of <math>A</math>. Further, their intersection is exactly <math>A</math>.

Localization at a prime ideal - Revision history

Vipul: 1 revision

Vipul at 15:57, 30 June 2007