Unique factorization domain

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

1. Every nonzero nonunit element can be written as a finite product of irreducible elements , and
2. This factorization is unique up to reordering and replacing factors by associates .

In a UFD, irreducible and prime elements coincide, which makes divisibility behave like the integers. Many polynomial rings over UFDs are again UFDs, enabling “induction on variables” arguments in commutative algebra.

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).