Prime numbers and prime ideals

Forward. If $p$ is prime and $ab\in(p)$, then $p\mid ab$, so $p\mid a$ or $p\mid b$ (Euclid's lemma), hence $a\in(p)$ or $b\in(p)$. So $(p)$ is prime.

Converse. If $(p)$ is prime and $p\mid ab$, then $ab\in(p)$ so $a\in(p)$ or $b\in(p)$: $p\mid a$ or $p\mid b$. This is the definition of primality.

Conclusion. Prime numbers correspond exactly to prime ideals of $\mathbb{Z}$.