Example of ideal which is not primary, but its radical is prime.

Question

I want to find an example of ideal $Q$ such that $\sqrt{Q}$ is prime, but $Q$ is not primary.

It is clear that our domain would not be PID because $\sqrt{Q}$ should not be maximal ideal.

Answer 1

Take $Q=(xy,y^2)\subseteq k[x,y]$. Then $\sqrt{Q}=(xy,y)=(y)$, which is prime, and $Q$ is not primary because $xy\in Q$, $y\notin Q$, but $x^n\notin Q$ for all $n\ge1$.