I want to find the $\DeclareMathOperator{\Spec}{Spec} \newcommand{\Z}{\mathbb Z} \Spec(\Z \times \Z)$ , I know the $\Spec(\Z)=\{0,(2),(3), \ldots \}$. If there is any relation between $\Spec(\Z \times \Z)$ and the $\Spec(\Z)$. I can find some of them like $$((0,p)),((p,0)),((p,q))$$ where $p$ and $q$ are prime numbers. I am not sure about the rest if there are, how can I argue that ?
Any suggestions will be appreciated.
Hint: Also $I=\{(n,0) \mid n\in \mathbb{Z}\}$ is a prime ideal, since the quotient $(\mathbb{Z}\times \mathbb{Z})/I$ is isomorphic to $\mathbb{Z}$, which is an integral domain. And note, that $((0,0))$ is not a prime ideal, since $\mathbb{Z}\times \mathbb{Z}$ is not an integral domain.
So we have the prime ideals $\mathbb{Z}\times p\mathbb{Z}$, $q\mathbb{Z}\times \mathbb{Z}$, $0\times \mathbb{Z}$ and $\mathbb{Z}\times 0$, for primes $p,q$.