Radical of height $2$ ideal in polynomial ring of two variables

Question

Let $J$ be an ideal of $\mathbb C[X,Y]$ of height $2$. Is it true that $\sqrt J$ is maximal ?

Answer 1

No. For instance, let $M$ and $M'$ be two different maximal ideals and $J=M\cap M'$. Then $J$ is radical and height 2 (the only primes containing it are $M$ and $M'$) but not maximal.

More generally, radical ideals of a commutative ring $\mathbb{C}[x,y]$ correspond to closed subsets of $\operatorname{Spec} \mathbb{C}[x,y]$, and height 2 radical ideals correspond to closed subsets all of whose points are closed. Such closed subsets are just finite sets of closed points. So any set of finitely many but more than one closed point gives a radical ideal which is height 2 but not maximal (namely, the intersection of the maximal ideals corresponding to the closed points).