Theorem 1

Let A be a Noetherian ring and M a finitely generated A-module. There exists a composition series $(M_i)_{0 \leq i \leq n}$ of M such that, for $0 \leq i \leq n - 1$, $M_i / M_{i+1}$ is isomorphic to $A / \mathfrak{p}_i$, where $\mathfrak{p}_i$ is a prime ideal of A.

Let $\mathscr{G}$ be the set of submodules of M which have a composition series with the property of the statement. As $\mathscr{G}$ is non-empty (for $\{0\}$ belongs to $\mathscr{G}$) and M is Noetherian, $\mathscr{G}$ has a maximal element N. If $M \neq N$, then $M/N \neq 0$ and hence $\operatorname{Ass}(M/N) \neq \varnothing$ (no. 1, Corollary 1 to Proposition 2); $M/N$ therefore contains a submodule $N'/N$ isomorphic to an A-module of the form $A/\mathfrak{p}$, where $\mathfrak{p}$ is prime; then by definition $N' \in \mathscr{G}$, which contradicts the maximal character of N. Then necessarily $N = M$.