Let M be a R graded module with $M= \oplus M_i$. If M is noetherian then $M_i=0$ for i << 0. My question is this, isn't $M_i = 0$ for all i >> 0 as well? If $(M_{n_i})_{i} \neq 0, n_i > 0$ then $M_{n_1} \nsubseteq M_{n_1} \oplus. .. M_{n_2} \nsubseteq M_{n_1} \oplus. .. M_{n_2} \oplus. .. M_{n_3} \nsubseteq. ..$ isn't a contradiction with ACC rule?
• Is $R$ a $\mathbb{N}$-graded ring? Then is $M$ a graded $R$-module, compatible with $R$'s grading? If that's the case, then the various $M_i$ you are writing are not even $R$-modules, they are $R_0$-modules. – Karl Schwede Jun 17 '11 at 20:01
Pick your favorite noetherian graded ring $R$, and consider the free module $M=R$. Are you saying it must vanish in high degree?