Normal domain: Difference between revisions
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===Weaker properties=== | ===Weaker properties=== | ||
==Metaproperties== | |||
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In fact, the [[localization at a multiplicatively closed subset]] of a normal domain continues to be a normal domain. {{proofat|[[Normal domain is localization-closed]]}} | |||
Revision as of 23:13, 15 March 2008
This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
View other properties of integral domains | View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions
Definition
Symbol-free definition
An integral domain is said to be normal if it is integrally closed in its field of fractions.
Relation with other properties
Stronger properties
Weaker properties
Metaproperties
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
In fact, the localization at a multiplicatively closed subset of a normal domain continues to be a normal domain. For full proof, refer: Normal domain is localization-closed