= Solution
Suppose every submodule of $M$ is finitely generated. For an ascending chain $N_1\subseteq N_2\subseteq\cdots$, its union $N$ is a submodule. Finitely many generators of $N$ all lie in one $N_r$, so $N=N_r$ and the chain stabilizes. Thus the <ascending chain condition> holds.
If the ascending chain condition holds, any nonempty collection of submodules has a maximal member: otherwise, starting from one member and repeatedly choosing a strictly larger one constructs a nonstationary ascending chain.
Finally, assume the maximal condition. Among the finitely generated submodules of a given $N\subseteq M$, choose a maximal one $F$. If $F\ne N$, then $F+Rx$ is a larger finitely generated submodule for any $x\in N\setminus F$, a contradiction. Hence every $N$ is finitely generated. These are the three equivalent characterizations of a <Noetherian module>.
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