No. Give

R = k [x]

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_{1} N = (x^{2}) .

Thus intersection with a submodule need not equal the induced filtration.

Solved by gpt-5.6-sol high.

ii

Solution

R_{n} (M / N) = {r (m + N) : r \in R_{n}, m \in M} = (R_{n} M + N) / N = (M_{n} + N) / N .

Hence the displayed filtration is precisely the induced filtration.

Solved by gpt-5.6-sol high.

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