Unique factorization domain

In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units.
Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field.
Unique factorization domains appear in the following chain of class inclusions:
Definition.
Formally, a unique factorization domain is defined to be an integral domain "R" in which every non-zero element "x" of "R" can be written as a product (an empty product if "x" is a unit) of irreducible elements "p"i of "R" and a unit "u":
and this representation is unique in the following sense:
If "q"1, ..., "q""m" are irreducible elements of "R" and "w" is a unit such that
then "m" = "n", and there exists a bijective map "φ" : {1, ..., "n"} → {1, ..., "m"} such that "p""i" is associated to "q""φ"("i") for "i" ∈ {1, ..., "n"}.
The uniqueness part is usually hard to verify, which is why the following equivalent definition is useful:
Examples.
Most rings familiar from elementary mathematics are UFDs:
Properties.
Some concepts defined for integers can be generalized to UFDs:
Equivalent conditions for a ring to be a UFD.
A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given at the end). Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case, it is in fact a principal ideal domain.
In general, for an integral domain "A", the following conditions are equivalent:
In practice, (2) and (3) are the most useful conditions to check. For example, it follows immediately from (2) that a PID is a UFD, since every prime ideal is generated by a prime element in a PID.
For another example, consider a Noetherian integral domain in which every height one prime ideal is principal. Since every prime ideal has finite height, it contains height one prime ideal (induction on height) which is principal. By (2), the ring is a UFD.