Unique factorization domains

Definition. An integral domain $R$ is a UFD if every nonzero non-unit element can be written as a product of irreducibles, uniquely up to order and units.

Example UFD. $\mathbb{Z}$, $\mathbb{R}[X]$, $\mathbb{Z}[i]$ (Euclidean $\Rightarrow$ PID $\Rightarrow$ UFD).

Non-example. $\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5}:a,b\in\mathbb{Z}\}$. It is an integral domain (subring of $\mathbb{C}$). However, $6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5})$ gives two distinct factorizations into irreducibles:
- $2$, $3$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$ (via norm argument);
- $1\pm\sqrt{-5}$ are irreducible (norm $6$).

So $\mathbb{Z}[\sqrt{-5}]$ is not a UFD.

Conclusion. UFD is a strong property, satisfied by PIDs but not all integral domains.