---
tag: YQNZTQGP1X
kind: concept
name: Artinian module
msc: ["16P20", "13E10"]
specializes: [0NZ8JA25WK]
sources:
  - url: https://en.wikipedia.org/wiki/Artinian_module
same_as:
  wikidata: Q2919100
  wikipedia: Artinian module
---

## Summary

An Artinian module is a module in which every decreasing chain of submodules stops after finitely many steps.
It is the condition dual to that of a [[Noetherian module|AK1ZRCXR32]].
A module is both Artinian and Noetherian exactly when it has finite [[length|CZSCNZH8VY]].

## Explanations

### L1 human

In an Artinian module you cannot keep shrinking submodules forever.
A finite abelian group, seen as a $\Z$-module, is Artinian, since it has only finitely many subgroups.
The group $\Z$ is not Artinian, because $2\Z \supset 4\Z \supset 8\Z \supset \cdots$ never stops.
The two chain conditions are different: the group of all complex roots of unity of order a power of a prime $p$ is Artinian but not Noetherian as a $\Z$-module.