Statements (28)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
Ring (algebra) |
| gptkbp:category |
gptkb:Abstract_algebra
|
| gptkbp:contrastsWith |
gptkb:Artinian_ring
|
| gptkbp:defines |
A ring in which every ideal is finitely generated
A ring in which every ascending chain of ideals terminates |
| gptkbp:example |
A field
Polynomial ring over a field The ring of polynomials in infinitely many variables over a field Quotient ring of a Noetherian ring The ring of integers |
| gptkbp:firstAppearance |
1921
|
| gptkbp:generalizes |
gptkb:Dedekind_domain
gptkb:Principal_ideal_domain |
| gptkbp:hasProperty |
Every ideal is finitely generated
|
| https://www.w3.org/2000/01/rdf-schema#label |
Noetherian Ring
|
| gptkbp:implies |
Every finitely generated module is Noetherian
|
| gptkbp:introduced |
gptkb:Emmy_Noether
|
| gptkbp:namedAfter |
gptkb:Emmy_Noether
|
| gptkbp:property |
Every submodule of a finitely generated module is finitely generated
Every quotient of a Noetherian ring is Noetherian A finite direct product of Noetherian rings is Noetherian A polynomial ring over a Noetherian ring is Noetherian (Hilbert's basis theorem) |
| gptkbp:usedIn |
gptkb:Algebraic_geometry
gptkb:Commutative_algebra Module theory |
| gptkbp:bfsParent |
gptkb:Department_of_Mathematics,_UC_Berkeley
|
| gptkbp:bfsLayer |
7
|