# Definition:Noetherian Ring

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## 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 **Noetherian ring** as **noetherian**, dropping the capital **N**.

## Also see

- Results about
**Noetherian rings**can be found**here**.

## Source of Name

This entry was named for Emmy Noether.