# 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.

## Source of Name

This entry was named for Emmy Noether.