Let $A$ be a commutative ring and $N$ its nilradical. Assume $N$ is a prime ideal of $A$. Then it is known that Spec $A$ is irreducible for the Zariski topology. However the zero ideal may be not primary.
Is there a concrete example of this situation and what is the primary decomposition of the zero ideal? Does it decompose Spec $A$, even though it is irreducible?