Difference between revisions of "Noetherian ring"
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{{further|[[equivalence of definitions of Noetherian ring]]}} | {{further|[[equivalence of definitions of Noetherian ring]]}} | ||
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+ | ==Metaproperties== | ||
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+ | {| class="sortable" border="1" | ||
+ | ! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | ||
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+ | | [[satisfies metaproperty::polynomial-closed property of commutative unital rings]] || Yes || [[Noetherianness is polynomial-closed]] || Suppose <math>R</math> is a Noetherian ring. Then, the [[polynomial ring]] <math>R[x]</matH> is also a Noetherian ring. | ||
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+ | | [[satisfies metaproperty::quotient-closed property of commutative unital rings]] || Yes || [[Noetherianness is quotient-closed]] || Suppose <math>R</math> is a Noetherian ring and <math>I</math> is an [[ideal]] in <math>R</math>. Then, the [[quotient ring]] <math>R/I</math> is also a Noetherian ring. | ||
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+ | | [[dissatisfies metaproperty::subring-closed property of commutative unital rings]] || No || [[Noetherianness is not subring-closed]] || It is possible to have the following: <math>R</math> is a Noetherian ring and <math>S</math> is a (unital) subring of <math>R</math>, but <math>S</math> is not a Noetherian ring. For instance, consider any non-Noetherian [[integral domain]]; this is a subring of a [[field]], which is Noetherian. | ||
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+ | | [[satisfies metaproperty::localization-closed property of commutative unital rings]] || Yes || [[Noetherianness is localization-closed]] || Suppose <math>R</math> is a Noetherian ring and <math>S</math> is a multiplicatively closed subset of <math>R</math>. Then, the localization of <math>R</math> at <math>S</math> is also Noetherian. | ||
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+ | | [[satisfies metaproperty::finite direct product-closed property of commutative unital rings]] || Yes || [[Noetherianness is finite direct product-closed]] || Suppose <math>R_1, R_2,\dots,R_n</math> are Noetherian rings. Then, the direct product <math>R_1 \times R_2 \times \dots \times R_n</math> is also a Noetherian ring. | ||
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+ | | [[satisfies metaproperty::completion-closed property of commutative unital rings]] || Yes || [[Noetherianness is completion-closed]] || Suppose <math>R</matH> is a commutative unital ring and <math>M</math> is a [[maximal ideal]] in <math>R</math>. The completion of <math>R</math> at <math>M</math> is also Noetherian. | ||
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==Relation with other properties== | ==Relation with other properties== | ||
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Latest revision as of 03:01, 18 July 2013
This article is about a standard (though not very rudimentary) definition in commutative algebra. The article text may, however, contain more than just the basic definition
View a complete list of semi-basic definitions on this wiki
This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
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
Below are some equivalent definitions of Noetherian ring:
No. | Shorthand | A commutative unital ring is termed Noetherian if ... | A commutative unital ring is termed Noetherian if ... |
---|---|---|---|
1 | ascending chain condition on ideals | any ascending chain of ideals stabilizes after a finite length | arbitrary version: if is a well-ordered set and are ideals such that for , there exists some such that for all . countable chain version: If is an ascending chain of ideals, then there exists such that for all . |
2 | finite generation of ideals | every ideal in the ring is finitely generated. | for every ideal in , there exist elements such that is the ideal generated by . |
3 | finite generation of prime ideals | every prime ideal in the ring is finitely generated. | for every prime ideal in , there exist elements such that equals the ideal generated by . |
Equivalence of definitions
Further information: equivalence of definitions of Noetherian ring
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
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polynomial-closed property of commutative unital rings | Yes | Noetherianness is polynomial-closed | Suppose is a Noetherian ring. Then, the polynomial ring is also a Noetherian ring. |
quotient-closed property of commutative unital rings | Yes | Noetherianness is quotient-closed | Suppose is a Noetherian ring and is an ideal in . Then, the quotient ring is also a Noetherian ring. |
subring-closed property of commutative unital rings | No | Noetherianness is not subring-closed | It is possible to have the following: is a Noetherian ring and is a (unital) subring of , but is not a Noetherian ring. For instance, consider any non-Noetherian integral domain; this is a subring of a field, which is Noetherian. |
localization-closed property of commutative unital rings | Yes | Noetherianness is localization-closed | Suppose is a Noetherian ring and is a multiplicatively closed subset of . Then, the localization of at is also Noetherian. |
finite direct product-closed property of commutative unital rings | Yes | Noetherianness is finite direct product-closed | Suppose are Noetherian rings. Then, the direct product is also a Noetherian ring. |
completion-closed property of commutative unital rings | Yes | Noetherianness is completion-closed | Suppose is a commutative unital ring and is a maximal ideal in . The completion of at is also Noetherian. |
Relation with other properties
Conjunction with other properties
Conjunction | Other component of conjunction | Additional comments |
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Noetherian domain | integral domain | |
reduced Noetherian ring | reduced ring: it has no nonzero nilpotent elements. | |
Noetherian normal domain | normal domain | |
Noetherian unique factorization domain | unique factorization domain | |
local Noetherian ring | local ring | |
local Noetherian domain | local domain | |
zero-dimensional Noetherian ring | zero-dimensional ring: every prime ideal in it is a maximal ideal | |
one-dimensional Noetherian domain | one-dimensional domain: integral domain in which every nonzero prime ideal is maximal | |
finite-dimensional Noetherian ring | finite-dimensional ring: its Krull dimension is finite. |
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
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Polynomial ring over a field | for a field | click here | ||
Artinian ring | descending chain of ideals stabilizes eventually | Artinian implies Noetherian | Noetherian not implies Artinian | click here |
Principal ideal ring | every ideal is principal | principal ideal ring implies Noetherian | Noetherian not implies principal ideal ring | click here |
Dedekind domain | click here | |||
Cohen-Macaulay ring | click here | |||
Affine ring | click here |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
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Coherent ring | click here |