Lemma 1

Let A be a graded ring of type $\mathbf{Z}$, whose graduation $(A_n)$ is such that $A_n = 0$ for all $n < 0$ or $A_n = 0$ for all $n > 0$. Let M be a graded A-module of type $\mathbf{Z}$. For M to be a Noetherian A-module, it is necessary and sufficient that every graded submodule of M be finitely generated.

As $n \mapsto -n$ is an automorphism of the group $\mathbf{Z}$, we may restrict our attention to the case $A_n = 0$ for all $n > 0$. Let A' and M' denote the ring A and the module M with the filtrations associated with their respective graduations (no. 1, Example 1), which are exhaustive and separated; the hypothesis on A implies that A' is discrete and hence complete. If E is a sub-A-module of M, the filtered A'-module E' obtained by giving E the induced filtration is Hausdorff and its filtration is exhaustive; moreover $\operatorname{gr}(E')$ is identified with a graded sub-A-module of $M = \operatorname{gr}(M')$ and hence is finitely generated by hypothesis. The conclusion then follows from Corollary 1 to Proposition 12 of no. 9.