Equivalence of Definitions of Noetherian Module

Theorem
Let $A$ be a commutative ring with unity.

Let $M$ be an $A$-module.

Definition 1 implies Definition 2
Assume $N_1 \subseteq N_2 \subseteq N_3 \subseteq \cdots$ is an ascending chain of submodules of $M$.

By Definition 1 $N := \bigcup_{i \mathop = 1}^\infty N_i$ is a finitely generated submodule of $M$.

Suppose that $N$ is generated by $a_1, \dotsc, a_k \in N$.

For all $i \in \set {1, \dots, k}$, there is some $j_i \in \N$, such that $a_i \in N_{j_i}$.

For $j := \max \set {j_1, \dots, j_k}$, we have:
 * $a_1, \dotsc, a_k \in N_j$

Hence $N_j = N$.

This shows, that the ascending chain stabilizes, as desired.

Definition 2 implies Definition 1
Let $N$ be a submodule of $M$.

$N$ is not finitely generated.

Any finitely generated submodule of $N$ is not equal to $N$.

So we can inductively choose a sequence:
 * $a_i \in N \setminus \sequence {a_1, \dots, a_{i - 1} }$.

The chain:
 * $\sequence {a_1} \subsetneq \sequence {a_1, a_2} \subsetneq \sequence {a_1, a_2, a_3} \subsetneq \dotsb$

is strictly increasing.

This contradicts Definition 2.

Thus $N$ is finitely generated.

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

Also see

 * Equivalence of Definitions of Noetherian Ring