Noetherian module
E29376
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Noetherian module canonical | 4 |
| Noetherian condition | 1 |
| Noetherian modules | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T228993 — 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 module Context triple: [Emmy Noether, notableWork, Noetherian module]
-
A.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
B.
Minkowski functional
The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
-
C.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
-
D.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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E.
von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
- 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 module Target entity description: 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.
-
A.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
B.
Minkowski functional
The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
-
C.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
-
D.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
E.
von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure property
ⓘ
mathematical concept ⓘ module theory concept ⓘ |
| characterizedBy |
ascending chain condition on submodules
ⓘ
finiteness of generation of all submodules ⓘ |
| closedUnder |
finite direct sums
ⓘ
taking quotients ⓘ taking submodules ⓘ |
| contrastsWith | Artinian module ⓘ |
| definedOver | ring ⓘ |
| enables |
finiteness arguments in module theory
ⓘ
induction on submodules ⓘ |
| equivalentCondition |
every increasing sequence of submodules becomes stationary
ⓘ
every nonempty family of submodules has a maximal element under inclusion ⓘ every submodule is the sum of finitely many cyclic submodules ⓘ |
| equivalentTo | module in which every submodule is finitely generated ⓘ |
| field |
abstract algebra
ⓘ
module theory ⓘ ring theory ⓘ |
| generalizes |
Noetherian rings
ⓘ
surface form:
Noetherian ring
|
| hasDualConcept | Artinian module ⓘ |
| hasExample |
finite-dimensional vector space over a field
ⓘ
finitely generated abelian group ⓘ finitely generated module over a Noetherian ring ⓘ |
| hasImportance |
allows reduction to finitely generated substructures
ⓘ
controls complexity of submodule structure ⓘ |
| hasNonExample |
direct sum of countably many copies of a nonzero module
ⓘ
polynomial ring in infinitely many variables over a field as a module over itself ⓘ |
| hasProperty |
every ascending chain of submodules stabilizes
ⓘ
satisfies ascending chain condition on submodules ⓘ |
| implies |
every nonempty set of submodules has a maximal element
ⓘ
every submodule is finitely generated ⓘ finitely generated over a Noetherian ring is Noetherian ⓘ |
| isGeneralizationOf | Noetherian abelian group ⓘ |
| mayBe | finitely generated over a Noetherian ring ⓘ |
| namedAfter | Emmy Noether ⓘ |
| notClosedUnder | arbitrary direct sums ⓘ |
| over |
Noetherian ring
ⓘ
commutative ring ⓘ noncommutative ring ⓘ |
| relatedTo |
Hilbert basis theorem
ⓘ
Krull dimension ⓘ associated primes ⓘ primary decomposition ⓘ |
| studiedIn |
Noetherian rings
ⓘ
surface form:
Noetherian ring theory
|
| usedIn |
algebraic geometry
ⓘ
commutative algebra ⓘ homological algebra ⓘ representation theory ⓘ |
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 module Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Noetherian condition
subject surface form:
Emmy Noether
this entity surface form:
Noetherian modules
subject surface form:
Noetherian ring