Normal domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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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

• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Dedekind domain

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.
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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

Ring of integer-valued polynomials

The ring of integer-valued polynomials over a normal domain is again a normal domain. For full proof, refer: Ring of integer-valued polynomials over normal domain is normal