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

• Equivalence of Definitions of Noetherian Module
• Short Exact Sequence Condition of Noetherian Modules
• Definition:Noetherian Ring

• Results about Noetherian modules can be found here.

Source of Name

This entry was named for Emmy Noether.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Noetherian module