 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[Yes.] Scalar multiplication in the <quotient module> gives directly
$$
R_n(M/N)=\{r(m+N):r\in R_n,m\in M\}
=(R_nM+N)/N=(M_n+N)/N.
$$
\b[Hence the displayed filtration is precisely the induced filtration.]

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