Unique factorization domain

In mathematics, a unique factorization domain (UFD) is a commutative ring, which is an integral domain, and 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, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki.

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 p_i of R and a unit u:

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

Most rings familiar from elementary mathematics are UFDs:

Non-example:

Some concepts defined for integers can be generalized to UFDs:

A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given below). 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.

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