Noetherian module

generated byleft noetherianright noetherian

A (left or right) module $M$ over a ring $R$ is said to be Noetherianif 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 non-commutative 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 aNoetherian topological space (http://planetmath.org/NoetherianTopologicalSpace).