Localization (commutative algebra): Difference between revisions

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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|IR}}, 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>