Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 101 / 5 / a

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- Solution a

Solution

Put

g = y^{2} - z w

. Every generator of

I

is divisible by

x^{2}

and belongs to

(y, z)^{2}

, while the displayed decomposition in the next part will show that

I = (x) \cap (y, z) .

Its two minimal primes, and hence its isolated associated primes, are

p_{1} = (x), p_{2} = (y, z) .

The corresponding isolated primary components are

q_{1} = (x^{2}), q_{2} = (y, z)^{2} = (y^{2}, yz, z^{2}) .

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