Proposition 2

Let $A$ be a torsion-free commutative group, $A$ a graded Noetherian ring of type $A$ and $M$ a graded finitely generated $A$-module of type $A$. There exists a composition series $(\mathbf{M}_i)_{0 \leq i \leq n}$ consisting of graded submodules of $M$ such that for $0 \leq i \leq n - 1$ the graded module $\mathbf{M}_i / \mathbf{M}_{i+1}$ is isomorphic to a shifted graded module $(A/p_i)(k_i)$, where $p_i$ is a graded prime ideal of $A$ and $k_i \in A$.

It is sufficient to retrace the argument of § 1, no. 4, Theorem 1 taking on this occasion $\mathscr{G}$ to be the set of graded submodules of $M$ with a composition series with the properties of the statement; we conclude using the Corollary to Proposition 1.