Unique factorization domain
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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 ... pn
and this representation is unique in the following sense: if q1,...,qm are irreducible elements of R such that
- x = q1 q2 ... qm,
then m = n and there exists a bijective map φ : {1,...,n} -> {1,...,n} such that pi is associated to qφ(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,...,Xn]] 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 is a counterexample:
- 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).
Most factor rings of a polynomial ring are not UFDs. Here is an example:
- Let R be any commutative ring. Then R[X,Y,Z,W]/(XY-ZW) is not a UFD. It is clear that X, Y, Z, and W are all irreducibles, so the element XY=ZW has two factorizations into irreducible elements.
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, ..., Xn] as well as K[X1, ..., Xn] (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.fr:Anneau factoriel