# Noetherian module

From Encyclopedia of Mathematics

A module for which every submodule has a finite system of generators. Equivalent conditions are: the ascending chain condition for submodules (every strictly ascending chain of submodules breaks off after finitely many terms); every non-empty set of submodules ordered by inclusion contains a maximal element. Submodules and quotient modules of a Noetherian module are Noetherian. If, in an exact sequence

$$0\to M'\to M\to M''\to0,$$

$M'$ and $M''$ are Noetherian, then so is $M$. A module over a Noetherian ring is Noetherian if and only if it is finitely generated. A module has a composition series if and only if it is both Artinian and Noetherian.

#### References

[1] | S. Lang, "Algebra" , Addison-Wesley (1974) |

**How to Cite This Entry:**

Noetherian module.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Noetherian_module&oldid=34020

This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article