TITLE: What is an example of a radical of sum of ideals not being equal to the sum of radicals?
QUESTION [8 upvotes]: What is an example of the radical of a sum of ideals not equal
to sum of the radical of the ideals?

REPLY [7 votes]: 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]