A commutative integral domain is a unique factorization domain (UFD for short) if every non-unit has a factorization as product of irreducible non-units and this decomposition is unique up to renumbering and rescaling the irreducibles by units.
Put differently: is a UFD precisely when the multiplicative monoid of nonzero principal ideals of (which is isomorphic to the monoid , where denotes the group of units) is a commutative monoid freely generated by irreducible principal ideals. It follows that if is the field of fractions of , then the quotient group is an abelian group that is freely generated by the set of cosets with ranging over irreducible elements. As a side remark, we observe that in this circumstance the exact sequence
splits and there is an isomorphism of abelian groups.
The ring of (rational) integers is a UFD.
If is a UFD, then its polynomial ring is also a UFD.
If is a UFD, then so is any localization .
If is a UFD and all of its localizations at primes are regular local rings, then the ring of formal power series is a UFD.
For a positive integer, the ring of integers in is a UFD iff is a Heegner number, namely one of the numbers .
If is not integrally closed, then it is not a UFD.
As noted above, a UFD is necessarily integrally closed.
The lattice of principal ideals under the inclusion order is a distributive lattice.