Unique factorization domain

A unique factorization domain (UFD) is an integral domain $R$ such that:

every nonzero nonunit is a finite product of irreducible elements; and
any two such factorizations differ only by reordering the factors and replacing factors by associates.

Remarks

In a UFD, every irreducible element is prime. If $R$ is a UFD, then $R[x]$ is a UFD; by iteration, so is $R[x_1,\dots,x_n]$ .

Examples

$\mathbb{Z}$ is a UFD.
If $k$ is a field, then $k[x,y]$ is a UFD.
$\mathbb{Z}[\sqrt{-5}]$ is not a UFD (e.g. $6$ has essentially different factorizations).