# Definition:Noetherian Module

## Definition

Let $A$ be a commutative ring with unity.

Let $M$ be an $A$-module.

### Definition 1

$M$ is a Noetherian module if and only if every submodule of $M$ is finitely generated.

### Definition 2

$M$ is a Noetherian module if and only if it satisfies the ascending chain condition on submodules.

### Definition 3

$M$ is a Noetherian module if and only if it satisfies the maximal condition on submodules.

## Also known as

Some sources render the term as noetherian, dropping the capital N.

## Also see

• Results about Noetherian modules can be found here.

## Source of Name

This entry was named for Emmy Noether.