Proposition 2

An A-module M is Noetherian if and only if every submodule of M is finitely generated.

Begin by assuming that every submodule of M is finitely generated. Let $(P_n)_{n\in\mathbf{N}}$ be an increasing sequence of submodules of M, and let P be its union. This is a submodule of M. By assumption, there exists a finite subset F of M generating the module P; let $n\in \mathbf{N}$ be an integer such that $F\subset P_n$. We then have $P_n= P$, and the sequence $(P_n)_{n\in\mathbf{N}}$ is stationary. This proves that the module M is Noetherian.

The converse is a consequence of the following more precise statement.