# Noetherian module

A (left or right) module $M$ over a ring $R$ is said to be Noetherian if the following equivalent conditions hold:

1. 1.

Every submodule of $M$ is finitely generated over $R$.

2. 2.

The ascending chain condition holds on submodules.

3. 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