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Statements (29)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:category |
abstract algebra
ring theory |
| gptkbp:contrastsWith |
gptkb:Artinian_rings
|
| gptkbp:defines |
A ring in which every ideal is finitely generated
A ring in which every ascending chain of ideals stabilizes |
| gptkbp:example |
fields
polynomial rings over a field the ring of integers the ring of polynomials in infinitely many variables over a field |
| gptkbp:field |
gptkb:algebra
|
| https://www.w3.org/2000/01/rdf-schema#label |
Noetherian rings
|
| gptkbp:implies |
every finitely generated module is Noetherian
|
| gptkbp:introduced |
gptkb:Emmy_Noether
|
| gptkbp:namedAfter |
gptkb:Emmy_Noether
|
| gptkbp:property |
Noetherian property is not preserved under infinite direct sums
every quotient of a Noetherian ring is Noetherian Hilbert's basis theorem: if R is Noetherian, so is R[x] finite direct products of Noetherian rings are Noetherian every localization of a Noetherian ring is Noetherian subrings of Noetherian rings need not be Noetherian Noetherian property is preserved under finite ring extensions every submodule of a finitely generated module is finitely generated |
| gptkbp:relatedTo |
gptkb:Hilbert's_basis_theorem
|
| gptkbp:usedIn |
gptkb:algebraic_geometry
gptkb:commutative_algebra homological algebra |
| gptkbp:bfsParent |
gptkb:Emmy_Noether
|
| gptkbp:bfsLayer |
4
|