Unique factorization domain

In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. Rings which are UFDs are sometimes called factorial, following the terminology of Bourbaki.

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

x = p₁ p₂ ... p_n

and this representation is unique in the following sense: if q₁,...,q_m are irreducible elements of R such that

x = q₁ q₂ ... q_m,

then m = n and there exists a bijective map φ : {1,...,n} -> {1,...,n} such that p_i is associated to q_φ(i) for i = 1, ..., n.

The uniqueness part is sometimes 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 non-unit can be written as a product of prime elements of R.

Examples

Most rings familiar from elementary mathematics are UFD's:

• the integers. This is the fundamental theorem of arithmetic.
• any field; this includes the fields of rational numbers, real numbers, and complex numbers.
• Rings of polynomials with coefficients in a field.

Here are some more exotic examples of UFDs:

• The Gaussian integers, Z[i].
• The formal power series ring K[[X₁,...,X_n]] over a field K.
• The ring of functions in n complex variables holomorphic at the origin is a UFD.

Despite these examples, very few integral domains are UFDs. Here are a few counterexamples:

• The ring

K[X,Y] / (Y² - X² + 1),

where K is a field. Then Y² factors as YY and as (X − 1)(X + 1). Most factor rings of a polynomial ring are not UFDs.

• The ring of all complex numbers of the form a + b √ −5, where a and b are integers. Then 6 factors as both (2)(3) and as (1 + √ −5) (1 − √ −5).

Properties

Additional examples of UFDs can be constructed as follows:

• All principal ideal domains are UFDs.

• If R is a UFD, then so is the polynomial ring R[X]. By induction, we can show that the polynomial rings Z[X₁, ..., X_n] as well as K[X₁, ..., X_n] (K a field) are UFD's. (Any polynomial ring with more than one variable is an example of a UFD that is not a principal ideal domain.)

Some concepts defined for integers can be generalized to UFDs:

• In UFD's, every irreducible element is prime. (In any integral domain, every prime element is irreducible, but the converse does not always hold.)

• Any two (or finitely many) elements of a UFD have a greatest common divisor and a least common multiple. Here, a greatest common divisor of a and b is an element d which divides both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are associated.

Equivalent conditions for a ring to be a UFD

Under some circumstances, it is possible to give 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.

• An integral domain is a UFD if and only if the ascending chain condition holds for principal ideals, and any two elements of A have a least common multiple.

• A ring is a UFD if and only if its class group is zero.