 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-101/4/c/i/solution

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.

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