Prime ideals in $\mathbb{Z}$

Prime ideals. $n\mathbb{Z}$ is prime iff $\mathbb{Z}/n\mathbb{Z}$ is an integral domain, iff $n$ is prime or $n=0$ (since $\mathbb{Z}/0\mathbb{Z}\cong\mathbb{Z}$ is an integral domain).

Maximal ideals. $n\mathbb{Z}$ is maximal iff $\mathbb{Z}/n\mathbb{Z}$ is a field, iff $n$ is prime.

Conclusion.
- Prime ideals: $(0)$ and $(p)$ for $p$ prime.
- Maximal ideals: $(p)$ for $p$ prime (not $(0)$, since $\mathbb{Z}$ is not a field).