Definition:Noetherian Ring

Definition

Definition 1

A commutative ring with unity $A$ is Noetherian if and only if every ideal of $A$ is finitely generated.

Definition 2

A commutative ring with unity $A$ is Noetherian if and only if it satisfies the ascending chain condition on ideals.

Definition 3

A commutative ring with unity $A$ is Noetherian if and only if it satisfies the maximal condition on ideals.

Definition 4

A commutative ring with unity $A$ is Noetherian if and only if it is Noetherian as an $A$-module.

Also known as

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

Also see

• Equivalence of Definitions of Noetherian Ring
• Definition:Noetherian Module

Source of Name

This entry was named for Emmy Noether.