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

Say $N$ is generated by $a_1, \dots, a_k \in N$.

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

For $j := \max\{j_1, \dots, j_k\}$, we have $a_1, \dots, a_k \in N_{j}$, hence $N_j = N$.

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

Definition 2 implies Definition 1
Let $N$ be a submodule of $M$. Assume, that $N$ is not finitely generated.

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

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

The chain $ \langle a_1 \rangle \subsetneq \langle a_1, a_2 \rangle \subsetneq \langle a_1, a_2, a_3 \rangle \subsetneq \dots $ 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