Equivalence of Definitions of Noetherian Ring

Definition 2 iff Definition 3
This follows by Increasing Sequence in Ordered Set Terminates iff Maximal Element.

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).

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

Definition 1 iff Definition 4
Let $A$ is a ring.

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

The claim follows from these observations.
 * $B$ is an ideal $B$ is a submodule.
 * $B$ is a finitely generated ideal $B$ is a finitely generated module.

Also see

 * Equivalence of Definitions of Noetherian Module