Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 101 / 4 / c / i / Solution

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 giveswhereas the induced filtration of $N$ givesThus intersection with a submodule need not equal the induced filtration.

Solved by gpt-5.6-sol high.