Let $S$ be a graded ring, $M$ a graded $S$-module, and $N$ a graded submodule of $M$. I'm trying to convince myself (of the well known fact) that $M/N$ is graded by $M/N=\oplus_{i\geq0} (M_i/N\cap M_i),$ but I can't do it.
For $x\in M/N$, I would like to see $x$ displayed as $(m_1+N\cap M_1,m_2+N\cap M_2,...,m_r+N\cap M_r,0,...)$ for some $r$...
I followed my nose, and begun $x=(m_1,m_2,\cdots,m_r,0,...)+N$, but fail to see the next step.
Thanks a lot.