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What is an example of the radical of a sum of ideals not equal to sum of the radical of the ideals?

ast
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1 Answers1

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In $\mathbb Q[x,y]$ we have: $$\sqrt {(x)+(x-y^2)}=\sqrt{(x,y^2)}=(x,y)\neq \sqrt {(x)} +\sqrt {(x-y^2)}=(x)+(x-y^2)=(x,y^2)$$

[Optional remark: If you know the basic dictionary relating commutative rings to affine schemes, you will note that this is just an example of the phenomenon that the intersection of two reduced subschemes of some affine scheme needn't be reduced]

Georges Elencwajg
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