Noetherian module
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generated by \PMlinkescapephraseleft noetherian^{} \PMlinkescapephraseright noetherian
A (left or right) module $M$ over a ring $R$ is said to be Noetherian if the following equivalent^{} conditions hold:

1.
Every submodule^{} of $M$ is finitely generated^{} over $R$.

2.
The ascending chain condition^{} holds on submodules.

3.
Every nonempty family of submodules has a maximal element^{}.
For example, the $\mathbb{Z}$module $\mathbb{Q}$ is not Noetherian, as it is not finitely generated, but the $\mathbb{Z}$module $\mathbb{Z}$ is Noetherian, as every submodule is generated by a single element.
Observe that changing the ring can change whether a module is Noetherian or not: for example, the $\mathbb{Q}$module $\mathbb{Q}$ is Noetherian, since it is simple (http://planetmath.org/SimpleModule) (has no nontrivial submodules).
There is also a notion of Noetherian for rings (http://planetmath.org/Noetherian): a ring is left Noetherian if it is Noetherian as a left module over itself, and right Noetherian if it is Noetherian as a right module over itself. For noncommutative rings, these two notions can differ.
The corresponding property for groups is usually called the maximal condition.
Finally, there is the somewhat related notion of a Noetherian topological space (http://planetmath.org/NoetherianTopologicalSpace).
Title  Noetherian module 
Canonical name  NoetherianModule 
Date of creation  20130322 11:44:57 
Last modified on  20130322 11:44:57 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  24 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 13E05 
Classification  msc 33C75 
Classification  msc 33E05 
Classification  msc 14J27 
Classification  msc 86A30 
Classification  msc 14H52 
Related topic  Noetherian 
Defines  Noetherian 
Defines  Noetherian left module 
Defines  Noetherian right module 
Defines  left Noetherian module 
Defines  right Noetherian module 