Noetherian ring (in the commutative case)
GPTKB entity
Statements (33)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
Noetherian ring
commutative ring |
| gptkbp:defines |
A commutative ring in which every ascending chain of ideals stabilizes
A commutative ring in which every ideal is finitely generated |
| gptkbp:example |
gptkb:Polynomial_ring_in_finitely_many_variables_over_a_field
gptkb:Polynomial_ring_in_infinitely_many_variables_over_a_field gptkb:The_ring_of_integers_Z Any field |
| https://www.w3.org/2000/01/rdf-schema#label |
Noetherian ring (in the commutative case)
|
| gptkbp:implies |
Artinian ring is Noetherian if and only if it is finite
Every finitely generated algebra over a Noetherian ring is Noetherian (Hilbert's basis theorem) |
| gptkbp:namedAfter |
gptkb:Emmy_Noether
|
| gptkbp:property |
Every ideal is finitely generated
Every Noetherian ring is a ring with Krull dimension A finitely generated module over a Noetherian ring is Noetherian A localization of a Noetherian ring is Noetherian Every submodule of a finitely generated module is finitely generated Every Noetherian ring is coherent Every quotient of a Noetherian ring is Noetherian Satisfies the ascending chain condition on ideals A subring of a Noetherian ring need not be Noetherian Every Noetherian ring is a ring with finitely many minimal prime ideals A finite direct product of Noetherian rings is Noetherian Every Noetherian ring is a ring with the ACC on ideals Every Noetherian ring is a ring with primary decomposition of ideals |
| gptkbp:relatedTo |
gptkb:Hilbert's_basis_theorem
gptkb:Artinian_ring gptkb:Krull_dimension gptkb:Ideal Module Primary decomposition |
| gptkbp:bfsParent |
gptkb:Artinian_ring
|
| gptkbp:bfsLayer |
6
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