Normal domain: Difference between revisions

Latest revision as of 01:11, 4 July 2012

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

An integral domain $R$ is said to be normal if it satisfies the following equivalent conditions:

$R$ is integrally closed in its field of fractions.
If $S$ is a subring of the field of fractions of $R$ that contains $R$ as a subgroup of finite index, then $S=R$ .

Equivalence of definitions

Further information: equivalence of definitions of normal domain

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
unique factorization domain	integral domain with unique factorization	unique factorization implies normal	normal domain not implies UFD	Template:Intermediate notion short
principal ideal domain	integral domain where every ideal is principal	(via UFD)	(via UFD)	click here
Euclidean domain	integral domain having a Euclidean norm	(via UFD)	(via UFD)	click here
Noetherian normal domain	normal domain that is also a Noetherian ring			click here
Dedekind domain	Noetherian, normal, and one-dimensional			click here

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal ring	integrally closed in its total quotient ring			click here

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: Normality is localization-closed

Polynomial-closedness

This property of integral domains is closed under taking polynomials, i.e., whenever an integral domain has this property, so does the polynomial ring in one variable over it.
View other polynomial-closed properties of integral domains OR view polynomial-closed properties of commutative unital rings

The polynomial ring in one variable over a normal domain is again normal. For full proof, refer: Normality is polynomial-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

@@ Line 3: / Line 3: @@
 ==Definition==
-===Symbol-free definition===
+An [[integral domain]] <math>R</math> is said to be '''normal''' if it satisfies the following equivalent conditions:
-An [[integral domain]] is said to be '''normal''' if it is [[integrally closed subring|integrally closed]] in its [[field of fractions]].
+# <math>R</math> is [[integrally closed subring|integrally closed]] in its [[field of fractions]].
+# If <math>S</math> is a subring of the field of fractions of <math>R</math> that contains <math>R</math> as a subgroup of finite index, then <math>S = R</math>.
+===Equivalence of definitions===
+{{further|[[equivalence of definitions of normal domain]]}}
 ==Relation with other properties==
@@ Line 11: / Line 16: @@
 ===Stronger properties===
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
-! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Weaker than::Unique factorization domain]] || [[integral domain]] with unique factorization || [[unique factorization implies normal]] || [[normal domain not implies UFD]] || {{intermediate notion short|normal domain|unique factorization domain}}
+| [[Weaker than::unique factorization domain]] || [[integral domain]] with unique factorization || [[unique factorization implies normal]] || [[normal domain not implies UFD]] || {{intermediate notion short|normal domain|unique factorization domain}}
 |-
-| [[Weaker than::Principal ideal domain]] || [[integral domain]] where every [[ideal]] is [[principal ideal|principal]] || (via UFD)||  (via UFD) || {{intermediate notions short|normal domain|principal ideal domain}}
+| [[Weaker than::principal ideal domain]] || [[integral domain]] where every [[ideal]] is [[principal ideal|principal]] || (via UFD)||  (via UFD) || {{intermediate notions short|normal domain|principal ideal domain}}
 |-
 | [[Weaker than::Euclidean domain]] || [[integral domain]] having a [[Euclidean norm]] || (via UFD) || (via UFD) || {{intermediate notions short|normal domain|Euclidean domain}}
@@ Line 27: / Line 32: @@
 ===Weaker properties===
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
-! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
+! Property !! Meaning !!Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
 | [[Stronger than::Normal ring]] || integrally closed in its [[total quotient ring]] || || || {{intermediate notions short|normal ring|normal domain}}