Unique factorization domain

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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:

Contents

Definition

Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero x of R can be written as a product (including an empty product) of irreducible elements pi of R and a unit u:

and this representation is unique in the following sense: If q1,...,qm 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 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:

Examples

Most rings familiar from elementary mathematics are UFDs:

Further examples of UFDs are:

  • The formal power series ring K[[X1,...,Xn]] 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[X1, ..., Xn] as well as K[X1, ..., Xn] (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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