exercise on radical ideals

Question

I know that given a commutative ring with unit the sum of two radical ideal is not a radical ideal. I would want to know if for example in the ring of polynomial in $n$ variables with coefficient in the field of complex numbers is true that given two radical ideal their sum is again a radical

Answer 1

No, $\langle y - x^2 \rangle + \langle y \rangle = \langle y, x^2 \rangle$ is a counterexample in $K[x,y]$ for any field $K$.