Let $R$ be a graded Noetherian ring and $M$ be f.g. graded $R$-module. What I want to show is that
- Primary decomposition of submodule of $M$ can be taken in terms of homogeneous modules. (In fact it is sufficient to show at decomposition of $0$.)
- It is well known that there is a finite filtration $0=M_0 \subseteq \cdots \subseteq M_n=M $ such that $M_i/M_{i-1} \cong R/P_i$ for some prime ideal $P_i$ (in fact, this $P_i$ is an associated prime of $M$.) Can we take these $M_i$ and $P_j$ homogeneous ones??
I could verify that in given conditions, every associated primes of $M$ is homogeneous.
Thank you for any helps.