Template:Localization-closed idp: Difference between revisions

Latest revision as of 00:31, 6 February 2009

Closure under taking localizations

This property of integral domains is closed under taking localizations: the localization at a multiplicatively closed subset of a commutative unital ring with this property, also has this property. In particular, the localization at a prime ideal, and the localization at a maximal ideal, have the property.
View other localization-closed properties of integral domains | View other localization-closed properties of commutative unital rings

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-===Localization-closedness===
+===Closure under taking localizations===
+{{quotation|''This [[property of integral domains]] is [[localization-closed property of commutative unital rings|closed under taking localizations]]: the [[localization at a multiplicatively closed subset]] of a [[commutative unital ring]] with this property, also has this property. In particular, the [[localization at a prime ideal]], and the [[localization at a maximal ideal]], have the property.''<br>[[:Category:Localization-closed properties of integral domains|View other localization-closed properties of integral domains]] <nowiki>|</nowiki> [[:Category:Localization-closed properties of commutative unital rings|View other localization-closed properties of commutative unital rings]]}}<includeonly>[[Category:Localization-closed properties of integral domains]][[Category:Localization-closed properties of commutative unital rings]]</includeonly>
-''This property of [[integral domain]]s is preserved under localizations, viz if an integral domain satisfies the property, so does its localization at every prime ideal''<includeonly>[[Category:Localization-closed properties of integral domains]][[Category:Localization-closed properties of commutative rings]]</includeonly><noinclude>[[Category: Integral domain metaproperty satisfaction templates]]</noinclude>