Prelims:

$p$ is prime$⟺⟨p⟩$ is a prime ideal

Prime elements are such that if they divide a product, then their divide one of the factors.

Irreducible elements are ones that always have a unit as a factor.

Proof:

1. Let $p$ be prime. Then if $p\mid ab$ we have it so either $p\mid a$ or $p\mid b$. For this case assume $p\mid a$.
2. So $\mathrm{\exists }k$ such that $p=ka$. We have to now show that if this is the case, either $k$ or $a$ has to be a unit.