Noetherian module

generated by left noetherian right 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 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 a Noetherian topological space (http://planetmath.org/NoetherianTopologicalSpace).

Title	Noetherian module
Canonical name	NoetherianModule
Date of creation	2013-03-22 11:44:57
Last modified on	2013-03-22 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