Noetherian rings
E157398
Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Noetherian ring | 5 |
| Noetherian rings canonical | 3 |
| Noetherian ring theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382735 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Noetherian rings Context triple: [Emmy Noether, knownFor, Noetherian rings]
-
A.
Noetherian induction
Noetherian induction is a proof technique used in mathematics to establish statements about structures satisfying the descending chain condition, generalizing ordinary mathematical induction.
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B.
Noetherian module
A Noetherian module is an algebraic structure in which every ascending chain of submodules stabilizes, ensuring that all submodules are finitely generated and enabling powerful finiteness arguments in ring and module theory.
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C.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
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D.
Noetherian space
A Noetherian space is a topological space in which every descending chain of closed subsets stabilizes, mirroring the finiteness conditions of Noetherian rings in algebra.
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E.
Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Noetherian rings Target entity description: Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
-
A.
Noetherian induction
Noetherian induction is a proof technique used in mathematics to establish statements about structures satisfying the descending chain condition, generalizing ordinary mathematical induction.
-
B.
Noetherian module
A Noetherian module is an algebraic structure in which every ascending chain of submodules stabilizes, ensuring that all submodules are finitely generated and enabling powerful finiteness arguments in ring and module theory.
-
C.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
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D.
Noetherian space
A Noetherian space is a topological space in which every descending chain of closed subsets stabilizes, mirroring the finiteness conditions of Noetherian rings in algebra.
-
E.
Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure property
ⓘ
ring-theoretic property ⓘ |
| fieldOfStudy |
commutative algebra
ⓘ
ring theory ⓘ |
| hasAbbreviation | ACC on ideals ⓘ |
| hasDefinition | a ring in which every ascending chain of ideals stabilizes ⓘ |
| hasEquivalentDefinition | a ring in which every ideal is finitely generated ⓘ |
| hasExample |
any Artinian ring
ⓘ
any field ⓘ any principal ideal domain ⓘ coordinate ring of an affine variety over a field ⓘ k[x_1,…,x_n] over a field k ⓘ the ring of integers Z ⓘ |
| hasNonExample |
polynomial ring in infinitely many variables over a field
ⓘ
ring of all polynomials in countably many variables over Z ⓘ |
| hasOppositeConcept | non-Noetherian ring ⓘ |
| hasProperty |
Hilbert basis theorem holds for polynomial extensions
ⓘ
Krull dimension is well-defined and finite for many important examples ⓘ Spec is quasi-compact ⓘ every finitely generated module has a finite composition series if it is Artinian as well ⓘ every ideal has a finite primary decomposition ⓘ every ideal has a finite set of generators ⓘ every ideal has only finitely many minimal prime ideals over it ⓘ every ideal is contained in a maximal ideal ⓘ every ideal is the intersection of primary ideals in a minimal primary decomposition ⓘ every nonempty set of ideals has a maximal element under inclusion ⓘ every nonzero module has an associated prime when the ring is Noetherian ⓘ every open subset of Spec is quasi-compact in the Zariski topology ⓘ every prime ideal is an intersection of primary ideals ⓘ every submodule of a finitely generated module is finitely generated ⓘ finite direct product of Noetherian rings is Noetherian ⓘ localization of a Noetherian ring is Noetherian ⓘ polynomial ring in finitely many variables over a Noetherian ring is Noetherian ⓘ quotient of a Noetherian ring is Noetherian ⓘ satisfies ascending chain condition on ideals ⓘ spectrum is a Noetherian topological space ⓘ submodules of finitely generated modules are finitely generated ⓘ |
| hasRelatedConcept |
Artinian ring
ⓘ
Krull dimension ⓘ Noetherian module ⓘ primary decomposition ⓘ |
| hasTheorem |
Hilbert basis theorem
ⓘ
surface form:
Hilbert basis theorem: if R is Noetherian then R[x] is Noetherian
Krull’s principal ideal theorem applies to Noetherian rings ⓘ Lasker–Noether theorem on primary decomposition ⓘ |
| isGeneralizationOf |
Dedekind domain
ⓘ
principal ideal domain ⓘ |
| isUsedIn |
algebraic geometry
ⓘ
algebraic number theory ⓘ homological algebra ⓘ module theory ⓘ |
| namedAfter | Emmy Noether ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Noetherian rings Description of subject: Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Noetherian ring
this entity surface form:
Noetherian ring theory
this entity surface form:
Noetherian ring
this entity surface form:
Noetherian ring
this entity surface form:
Noetherian ring
this entity surface form:
Noetherian ring
subject surface form:
Introduction to Commutative Algebra