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: 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 pi of R and a unit u: x = u p1 p2 ⋅⋅⋅ pn with n ≥ 0 and this representation is unique in the following sense: If q1, ..., qm are irreducible elements of R and w is a unit such that x = w q1 q2 ⋅⋅⋅ qm with m ≥ 0, then m = n, and there exists a bijective map φ : {1, ..., n} → {1, ..., m} such that pi is associated to qφ(i) for i ∈ {1, ..., n}. The uniqueness part is usually 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 element can be written as a product of a unit and prime elements of R. 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. If R is a UFD, then so is R[X], the ring of polynomials with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD (and in particular over a field or over the integers) is a UFD.

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