Noetherian module

Builds on Ring Module over a ring Submodule Ascending chain condition Finitely generated module

Definition

Let $R$ be a ring and let $M$ be a left module over $R$ . The module $M$ is Noetherian if the set of its submodules, ordered by inclusion, satisfies the ascending chain condition. 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; 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, every finitely generated module is Noetherian.

Reading path

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Ascending chain condition definition
Group concept
Module over a ring definition
Ring definition
Finitely generated module definition
Submodule definition
Noetherian module definition

Tag	`PNERH1096H`
URI	`m2://PNERH1096H`
Kind	definition
Writing	W0 stub
MSC	13E05, 16P40
Defines	Noetherian module
Uses	Ring, Module over a ring, Submodule, Ascending chain condition, Finitely generated module
Changed	2 November 2026

Noetherian module

Definition

Remarks

Reading path

Further reading

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