Proposition 6

Let $\Delta$ be an ordered commutative group, A a graded ring of type $\Delta$, $(A_\lambda)$ its graduation, M a graded A-module of type $\Delta$ and $(M_\lambda)$ its graduation. Suppose that the relation $A_\lambda \neq \{0\}$ implies $\lambda \geq 0$ and that there exists $\lambda_0$ such that $M_{\lambda_0} \neq \{0\}$ and $M_\lambda = \{0\}$ for $\lambda < \lambda_0$. Then, if $\mathfrak{I}_0 = \sum_{\lambda > 0} A_\lambda$, $\mathfrak{I}_0 M \neq M$.

Let $x$ be a non-zero element of $M_{\lambda_0}$; suppose that $x \in \mathfrak{I}_0 M$. Then $x = \sum_i a_i x_i$, where the $a_i$ are homogeneous elements $\neq 0$ of $\mathfrak{I}_0$ and the $x_i$ homogeneous elements $\neq 0$ of M with $\deg(x) = \deg(a_i) + \deg(x_i)$ for all $i$ (no. 2). But, as $\deg(a_i) > 0$, $\lambda_0 = \deg(a_i) + \deg(x_i) > \deg(x_i)$, which contradicts the hypothesis.