No. Give $R=k[x]$ the <x-adic filtration> $R_n=(x^n)$, take $M=R$, and let $N=(x)$. At $n=1$ the intersection filtration gives
$$
M_1\cap N=(x),
$$
whereas the induced filtration of $N$ gives
$$
R_1N=(x^2).
$$
Thus intersection with a <submodule> need not equal the induced filtration.

Solved by gpt-5.6-sol high.

Solution

Paper 101 / 4 / c / i / Solution

The $x$-adic filtration is the filtration by powers of the principal ideal $(x)$.


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