---
tag: PNERH1096H
kind: definition
name: Noetherian module
msc: ["13E05", "16P40"]
defines: AK1ZRCXR32
uses: [0894J980RK, 09VDS2W1E8, 0RGZ10DFPG, 0ZCAXY9563, 0XHM30WBVB]
sources:
  - url: https://en.wikipedia.org/wiki/Noetherian_module
---

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $M$ be a left [[module|09VDS2W1E8]] over $R$.
The module $M$ is *Noetherian* if the set of its [[submodules|0RGZ10DFPG]], ordered by inclusion, satisfies the [[ascending chain condition|0ZCAXY9563]].
That is, for every chain $N_1 \subseteq N_2 \subseteq \cdots$ of submodules of $M$, there is an index $k$ with $N_n = N_k$ for all $n \ge k$.

## Remarks

The following conditions are equivalent: $M$ is Noetherian; every submodule of $M$ is [[finitely generated|0XHM30WBVB]]; every nonempty set of submodules of $M$ has a maximal element for inclusion.
Example: $\Z$ as a $\Z$-module is Noetherian, because each submodule is $n\Z$ for some $n$, generated by one element.
Non-example: $\Q$ as a $\Z$-module is not Noetherian, because $\Z \subsetneq \frac{1}{2}\Z \subsetneq \frac{1}{4}\Z \subsetneq \cdots$ is a chain of submodules that never stops.
Submodules and quotients of Noetherian modules are Noetherian, and an extension of two Noetherian modules is Noetherian.
Over a [[Noetherian ring|0MVH4Y1CRZ]], every finitely generated module is Noetherian.

## Explanations

### L1 human

A Noetherian module is a module with no endless increasing chain of submodules.
Equivalently, each of its submodules can be described by finitely many generators.