# Equivalence of Definitions of Noetherian Ring

This article needs to be linked to other articles.In particular: throughoutYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## Theorem

The following definitions of the concept of **Noetherian Ring** are equivalent:

### 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.

## Proof

### Definition 2 iff Definition 3

This follows by Increasing Sequence in Ordered Set Terminates iff Maximal Element.

$\Box$

### Definition 2 implies Definition 1

Assume there is an ideal $I$ which is not finitely generated.

For any finite set $\set {a_1, \dotsc, a_n}$ where $n \in \N$, the generated ideal is not equal to $I$.

Consider the chain:

- $\sequence {a_1} \subset \sequence {a_1, a_2} \subset \cdots$

This chain does not satisfy the ascending chain condition (note that $I$ has infinitely many elements by assumption).

$\Box$

### Definition 1 implies Definition 2

Let there be a chain of ideals $I_1 \subset I_2 \subset \cdots$.

Then $\ds J = \bigcup_{n \mathop \ge 1} I_n$ is an ideal.

Let $J$ be finitely generated, by $\set {b_1, \dotsc b_m}$ for some $m \in \N$.

As the chain is ascending, there exists an ideal such that:

- $\set {b_1, \dotsc b_m} \subset I_k$

for some $k \in \N$.

It follows that:

- $I_k = \ideal {b_1, \dotsc, b_m}$

Hence:

- $\forall l \ge k: I_l = I_k$

$\Box$

### Definition 1 iff Definition 4

Let $A$ is a ring.

For any subset $B\subseteq A$, we have:

- $B$ is an ideal if and only if $B$ is a submodule.
- $B$ is a finitely generated ideal if and only if $B$ is a finitely generated module.

The claim follows from these observations.

$\Box$

This theorem requires a proof.In particular: another proof using Definition:Noetherian Module and definition 4You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |