Noetherian module: Difference between revisions
No edit summary |
m (4 revisions) |
||
| (One intermediate revision by the same user not shown) | |||
| Line 8: | Line 8: | ||
A [[module]] over a [[commutative unital ring]] is termed '''Noetherian''' if every submodule of it is finitely generated. | A [[module]] over a [[commutative unital ring]] is termed '''Noetherian''' if every submodule of it is finitely generated. | ||
==Relation with other properties== | |||
===Weaker properties=== | |||
* [[Finitely generated module]] | |||
===Incomparable properties=== | |||
* [[Artinian module]] | |||
==Facts== | ==Facts== | ||
| Line 13: | Line 23: | ||
* A ring is Noetherian as a module over itself if and only if it is a [[Noetherian ring]]. This is because ideals of a ring are precisely its submodules. | * A ring is Noetherian as a module over itself if and only if it is a [[Noetherian ring]]. This is because ideals of a ring are precisely its submodules. | ||
* Every [[finitely generated module]] over a Noetherian ring is a Noetherian module. | * Every [[finitely generated module]] over a Noetherian ring is a Noetherian module. | ||
==Metaproperties== | |||
{{submodule-closed}} | |||
Any submodule of a Noetherian module is Noetherian. | |||
{{finite-extension-closed}} | |||
If a module has a Noetherian submodule and the quotient module is Noetherian, the module itself is Noetherian. | |||
{{quotient-module-closed}} | |||
Any quotient module of a Noetherian module is Noetherian. | |||
Latest revision as of 16:27, 12 May 2008
This article defines a property of a module over a commutative unital ring
A commutative unital ring which satisfies this property as a module over itself, is termed a/an: Noetherian ring
Definition
Symbol-free definition
A module over a commutative unital ring is termed Noetherian if every submodule of it is finitely generated.
Relation with other properties
Weaker properties
Incomparable properties
Facts
- A ring is Noetherian as a module over itself if and only if it is a Noetherian ring. This is because ideals of a ring are precisely its submodules.
- Every finitely generated module over a Noetherian ring is a Noetherian module.
Metaproperties
Any submodule of a Noetherian module is Noetherian.
Template:Finite-extension-closed
If a module has a Noetherian submodule and the quotient module is Noetherian, the module itself is Noetherian.
Template:Quotient-module-closed
Any quotient module of a Noetherian module is Noetherian.