If $M$ is a graded $R$-module ($R$ a graded ring) $\bigoplus$$M$$_i$ and $N$ is a graded $R$-submodule of $M$, $\bigoplus$$N$$_i$, then how do we write the grading on the quotient module $M/N$? I have read that $M/N$ $=$ $\bigoplus$$(M_i/N_i$). But I have also come across a homegeneous component of $M/N$ being referred to as of the form $x$ $+$ $N$ for $x$ in some $M_i$, which suggests to me we can write $N/M$ $=$ $\bigoplus$($M_i$/$N$).
Any clarification would be really helpful!
$M_i/N$ doesn't make sense because in general $N$ is not a subset of $M_i$. However, you could use $M_i/(M_i\cap N)$. Notice though that this is just $M_i/N_i$!