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.
Aliases (2)
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 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 ring theory
→
|
| usedIn |
algebraic geometry
→
commutative algebra → homological algebra → representation theory → |
Referenced by (5)
| Subject (surface form when different) | Predicate |
|---|---|
|
Emmy Noether
("Noetherian modules")
→
|
knownFor |
|
Emmy Noether
("Noetherian condition")
→
|
notableIdea |
|
Emmy Noether
→
|
notableWork |
|
Noetherian induction
→
|
relatedConcept |
|
Hilbert basis theorem
→
|
relatedTo |