Noetherian module


generated by \PMlinkescapephraseleft noetherianPlanetmathPlanetmath \PMlinkescapephraseright noetherian

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

  1. 1.

    Every submoduleMathworldPlanetmath of M is finitely generatedMathworldPlanetmathPlanetmath over R.

  2. 2.

    The ascending chain conditionMathworldPlanetmathPlanetmathPlanetmath holds on submodules.

  3. 3.

    Every nonempty family of submodules has a maximal elementMathworldPlanetmath.

For example, the -module is not Noetherian, as it is not finitely generated, but the -module 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 -module is Noetherian, since it is simple ( (has no nontrivial submodules).

There is also a notion of Noetherian for rings ( 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 (

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