Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero non-unit x of R can be written as a product of irreducible elements of R:
The uniqueness part is sometimes hard to verify, which is why the following equivalent definition is useful: a unique factorization domain is an integral domain R in which every non-zero non-unit can be written as a product of prime elements of R.
All principal ideal domains are UFD's; this includes the integers, all fields, all polynomial rings K[X] where K is a field, and the Gaussian integers Z[i].
In general, if R is a UFD, then so is the polynomial ring R[X]. By induction, we therefore see that the polynomial rings Z[X1,...,Xn] as well as K[X1,...,Xn] (K a field) are UFD's.
The formal power series ring K[[X1,...,Xn]] over a field K is also a unique factorization domain.
The ring of functions in n complex variables holomorphic at the origin is a UFD.
Here is an example of an integral domain which is not a UFD: the ring of all complex numbers of the form a + b √ -5, where a and b are integers.
In UFD's, every irreducible element is prime. (Generally, in any integral domain, every prime element is irreducible.)
Any two (or finitely many) elements of a UFD have a greatest common divisor and a least common multiple. Here, a greatest common divisor of a and b is an element d which divides both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are associated.Examples
Properties