Unique factorization domain

{{#invoke:Hatnote|hatnote}}Template:Main other In mathematics, a unique factorization domain (UFD) is a commutative ring 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:

Commutative ringsintegral domainsintegrally closed domainsunique factorization domainsprincipal ideal domainsEuclidean domainsfields

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 (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.

Examples

Most rings familiar from elementary mathematics are UFDs:

Properties

Some concepts defined for integers can be generalized to UFDs:

Equivalent conditions for a ring to be a UFD

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.

There are also equivalent conditions for non-noetherian integral domains. Let A be an integral domain. Then the following are equivalent.

1. A is a UFD.
2. Every nonzero prime ideal of A contains a prime element. (Kaplansky)
3. A satisfies ascending chain condition on principal ideals (ACCP), and the localization S−1A is a UFD, where S is a multiplicatively closed subset of A generated by prime elements. (Nagata criterion)
4. A satisfies (ACCP) and every irreducible is prime.
5. A is atomic and every irreducible is prime.
6. A is a GCD domain (i.e., any two elements have a greatest common divisor) satisfying (ACCP).
7. A is a Schreier domain, and atomic.
8. A is a pre-Schreier domain and atomic.
9. A has a divisor theory in which every divisor is principal.
10. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.)
11. A is a Krull domain and every prime ideal of height 1 is principal.

In practice, (2) and (3) are the most useful conditions to check. For example, it follows immediately from (2) that a PID is a UFD, since, in a PID, every prime ideal is generated by a prime element.

For another example, consider a Noetherian integral domain in which every height one prime ideal is principal. Since every prime ideal has finite height, it contains height one prime ideal (induction on height), which is principal. By (2), the ring is a UFD.