Editing Localization (commutative algebra) (section)

== 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>

Many properties of ideals are either preserved by saturation and localization, or can be characterized by simpler properties of localization and saturation.
In what follows, {{mvar|S}} is a multiplicative set in a ring {{mvar|R}}, and {{mvar|I}} and {{mvar|J}} are ideals of {{mvar|R}}; the saturation of an ideal {{mvar|I}} by a multiplicative set {{mvar|S}} is denoted <math>\operatorname{sat}_S (I),</math> or, when the multiplicative set {{mvar|S}} is clear from the context, <math>\operatorname{sat}(I).</math> 
* <math>1 \in S^{-1}I \quad\iff\quad 1\in \operatorname{sat}(I) \quad\iff\quad S\cap I \neq \emptyset</math>
* <math>I \subseteq J \quad\ \implies \quad\ S^{-1}I \subseteq S^{-1}J \quad\ \text{and} \quad\ \operatorname{sat}(I)\subseteq \operatorname{sat}(J)</math><br>(this is not always true for [[strict subset|strict inclusions]])
* <math>S^{-1}(I \cap J) = S^{-1}I \cap  S^{-1}J,\qquad\, \operatorname{sat}(I \cap J) = \operatorname{sat}(I) \cap \operatorname{sat}(J)</math>
* <math>S^{-1}(I + J) = S^{-1}I + S^{-1}J,\qquad \operatorname{sat}(I + J) = \operatorname{sat}(I) + \operatorname{sat}(J)</math>
* <math>S^{-1}(I \cdot J) = S^{-1}I \cdot  S^{-1}J,\qquad\quad \operatorname{sat}(I \cdot J) = \operatorname{sat}(I) \cdot \operatorname{sat}(J)</math>
* If <math>\mathfrak p</math> is a [[prime ideal]] such that <math>\mathfrak p \cap S = \emptyset,</math> then <math>S^{-1}\mathfrak p</math> is a prime ideal and <math>\mathfrak p = \operatorname{sat}(\mathfrak p)</math>; if the intersection is nonempty, then <math>S^{-1}\mathfrak p = S^{-1}R</math> and <math>\operatorname{sat}(\mathfrak p)=R.</math>