In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements (or irreducible elements), analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki.
Note that unique factorization domains appear in the following chain of class inclusions:
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Definition
Formally, a unique factorization domain is defined to be an integral domain R in which every nonzero x of R can be written as a product (including an empty product) 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:
 All principal ideal domains, hence all Euclidean domains, are UFDs. In particular, the integers (also see fundamental theorem of arithmetic), the Gaussian integers and the Eisenstein integers are UFDs.
 Any field is trivially a UFD, since every nonzero element is a unit. Examples of fields include rational numbers, real numbers, and complex numbers.
 If R is a UFD, then so is R[x], the ring of polynomials with coefficients in R. A special case of this, due to the above, is that the polynomial ring over any field is a UFD.
 Every regular local ring is a UFD.
Further examples of UFDs are:
 The formal power series ring K[[X_{1},...,X_{n}]] over a field K (or more generally over a PID but not over a UFD).
 The ring of functions in a fixed number of complex variables holomorphic at the origin is a UFD.
 By induction one can show that the polynomial rings Z[X_{1}, ..., X_{n}] as well as K[X_{1}, ..., X_{n}] (K a field) are UFDs. (Any polynomial ring with more than one variable is an example of a UFD that is not a principal ideal domain.)
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