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

Table of contents
- Solution 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 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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