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}) .

b

Solution

Consider

q_{3} = (x^{3}, y^{2} - z w) .

It is

(x, y^{2} - z w)

-primary. A direct ideal-intersection calculation gives

q_{1} \cap q_{2} \cap q_{3} = x^{2} [(y, z)^{2} \cap (x, g)] = x^{2} [x (y, z)^{2} + g (y, z)] = (x^{3} y^{2}, x^{3} yz, x^{3} z^{2}, x^{2} y (y^{2} - z w), x^{2} z (y^{2} - z w)) = I .

This decomposition is irredundant and its radicals are

(x), (y, z), (x, y^{2} - z w) .

Therefore

Ass (k [x, y, z, w] / I) = {(x), (y, z), (x, y^{2} - z w)} .

In particular,

p_{3} = (x, y^{2} - z w)

is the embedded associated prime and the decomposition also confirms

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

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