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?

Pick your favorite noetherian graded ring $R$, and consider the free module $M=R$. Are you saying it must vanish in high degree? 

