I have problems to give an example of noetherian ring $R$ whose nilradical is prime and $(0)$ is not primary.
I can find a lot of examples when $(0)$ is not primary, for example $\mathbb{C}[x,y]/(x^2,xy), \mathbb{C}[x,y]/(xy)$ etc. But problems arise in nilradical. I know that nilradical is prime iff spectrum of this ring is irreducible, but I don't know if it could help. I would like any hint.