Equivalence of Definitions of Noetherian Ring

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

$\Box$

Definition 2 implies Definition 1

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

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

Consider the chain:

$\left\langle{a_1}\right\rangle \subset \left\langle{a_1, a_2}\right\rangle \subset \cdots$

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

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

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

$\left\{ {b_1, \dotsc b_m }\right\} \subset I_k$

for some $k \in \N$.

It follows that:

$I_k = \left\langle{b_1, \dotsc, b_m}\right\rangle$

Hence:

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