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Two classes of localizations are more commonly considered:
* The multiplicative set is the [[complement (set theory)|complement]] of a [[prime ideal]] <math>\mathfrak p</math> of a ring {{mvar|R}}. In this case, one speaks of the "localization at <math>\mathfrak p</math>", or "localization at a point". The resulting ring, denoted <math>R_\mathfrak p</math> is a [[local ring]], and is the algebraic analog of a [[germ (mathematics)#Ring of germs|ring of germs]].
* The multiplicative set consists of all powers of an element {{mvar|t}} of a ring {{mvar|R}}. The resulting ring is commonly denoted <math>R_t,</math> and its spectrum is the Zariski [[open set]] of the prime ideals that do not contain {{mvar|t}}. Thus the localization is the analog of the restriction of a topological space to a neighborhood of a point (every prime ideal has a [[neighborhood basis]] consisting of Zariski open sets of this form).
{{anchor|away from}}In [[number theory]] and [[algebraic topology]], when working over the ring <math>\Z</math> of [[integer]]s, one refers to a property relative to an integer {{mvar|n}} as a property true ''at'' {{mvar|n}} or ''away'' from {{mvar|n}}, depending on the localization that is considered. "'''Away from''' {{mvar|n}}" means that the property is considered after localization by the powers of {{mvar|n}}, and, if {{mvar|p}} is a [[prime number]], "at {{mvar|p}}" means that the property is considered after localization at the prime ideal <math>p\Z</math>. This terminology can be explained by the fact that, if {{mvar|p}} is prime, the nonzero prime ideals of the localization of <math>\Z</math> are either the [[singleton set]] {{math|{{mset|p}}}} or its complement in the set of prime numbers.
== Localization and saturation of ideals ==
Let {{mvar|S}} be a multiplicative set in a commutative ring {{mvar|R}}, and <math>j\colon R\to S^{-1}R</math> be the [[canonical ring]] homomorphism. Given an [[ideal (ring theory)|ideal]] {{mvar|I}} in {{mvar|R}}, let <math>S^{-1}I</math> the set of the fractions in <math>S^{-1}R</math> whose numerator is in {{mvar|I}}. This is an ideal of <math>S^{-1}R,</math> which is generated by {{math|''j''(''I'')}}, and called the ''localization'' of {{mvar|I}} by {{mvar|S}}.
The ''saturation'' of {{mvar|I}} by {{mvar|S}} is <math>j^{-1}(S^{-1}I);</math> it is an ideal of {{mvar|R}}, which can also defined as the set of the elements <math>r\in R</math> such that there exists <math>s\in S</math> with <math>sr\in I.</math>
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Let <math>R</math> be a [[commutative ring]], <math>S</math> be a [[multiplicative set]] in <math>R</math>, and <math>M</math> be an <math>R</math>-[[module (mathematics)|module]]. The '''localization of the module''' <math>M</math> by <math>S</math>, denoted <math>S^{-1}M</math>, is an <math>S^{-1}R</math>-module that is constructed exactly as the localization of <math>R</math>, except that the numerators of the fractions belong to <math>M</math>. That is, as a set, it consists of [[equivalence class]]es, denoted <math>\frac ms</math>, of pairs <math>(m,s)</math>, where <math>m\in M</math> and <math>s\in S,</math> and two pairs <math>(m,s)</math> and <math>(n,t)</math> are equivalent if there is an element <math>u</math> in <math>S</math> such that
:<math>u(sn-tm)=0.</math>
Addition and [[scalar multiplication]] are defined as for usual fractions (in the following formula, <math>r\in R,</math> <math>s,t\in S,</math> and <math>m,n\in M</math>):
:<math>\frac{m}{s} + \frac{n}{t} = \frac{tm+sn}{st},</math>
:<math>\frac rs \frac{m}{t} = \frac{r m}{st}.</math>
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