Unique factorization domain

In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.

Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field.

Unique factorization domains appear in the following chain of class inclusions:

rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields

Definition

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 of a unit u and zero or more irreducible elements p_i of R:

x = u p₁ p₂ ⋅⋅⋅ p_n with n ≥ 0

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

x = w q₁ q₂ ⋅⋅⋅ q_m with m ≥ 0,

then m = n, and there exists a

• All
  integers (also see Fundamental theorem of arithmetic), the Gaussian integers and the Eisenstein integers
  are UFDs.
• If R is a UFD, then so is R[X], the ring of polynomials with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD (and in particular over a field or over the integers) is a UFD.
• The formal power series ring K[[X₁, ..., X_n]] over a field K (or more generally over a regular UFD such as a PID) is a UFD. On the other hand, the formal power series ring over a UFD need not be a UFD, even if the UFD is local. For example, if R is the localization of k[x, y, z]/(x² + y³ + z⁷) at the prime ideal (x, y, z) then R is a local ring that is a UFD, but the formal power series ring R[[X]] over R is not a UFD.
• The Auslander–Buchsbaum theorem states that every regular local ring is a UFD.
• ${\displaystyle \mathbb {Z} \left[e^{\frac {2\pi i}{n}}\right]}$ is a UFD for all integers 1 ≤ n ≤ 22, but not for n = 23.
• Mori showed that if the completion of a
  localization
  of k[x, y, z]/(x² + y³ + z⁵) at the prime ideal (x, y, z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x, y, z]/(x² + y³ + z⁷) at the prime ideal (x, y, z) the local ring is a UFD but its completion is not.
• Let ${\displaystyle R}$ be a field of any characteristic other than 2. Klein and Nagata showed that the ring R[X₁, ..., X_n]/Q is a UFD whenever Q is a nonsingular quadratic form in the Xs and n is at least 5. When n = 4, the ring need not be a UFD. For example, R[X, Y, Z, W]/(XY − ZW) is not a UFD, because the element XY equals the element ZW so that XY and ZW are two different factorizations of the same element into irreducibles.
• The ring Q[x, y]/(x² + 2y² + 1) is a UFD, but the ring Q(i)[x, y]/(x² + 2y² + 1) is not. On the other hand, The ring Q[x, y]/(x² + y² − 1) is not a UFD, but the ring Q(i)[x, y]/(x² + y² − 1) is.
  coordinate ring R[X, Y, Z]/(X² + Y² + Z² − 1) of the 2-dimensional real sphere
  is a UFD, but the coordinate ring C[X, Y, Z]/(X² + Y² + Z² − 1) of the complex sphere is not.
• Suppose that the variables X_i are given weights w_i, and F(X₁, ..., X_n) is a homogeneous polynomial of weight w. Then if c is coprime to w and R is a UFD and either every finitely generated projective module over R is free or c is 1 mod w, the ring R[X₁, ..., X_n, Z]/(Z^c − F(X₁, ..., X_n)) is a UFD.^[3]

Non-examples

• The
  quadratic integer ring
  ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ of all complex numbers of the form ${\displaystyle a+b{\sqrt {-5}}}$, where a and b are integers, is not a UFD because 6 factors as both 2×3 and as ${\displaystyle \left(1+{\sqrt {-5}}\right)\left(1-{\sqrt {-5}}\right)}$. These truly are different factorizations, because the only units in this ring are 1 and −1; thus, none of 2, 3, ${\displaystyle 1+{\sqrt {-5}}}$, and ${\displaystyle 1-{\sqrt {-5}}}$ are associate. It is not hard to show that all four factors are irreducible as well, though this may not be obvious.^[4] See also Algebraic integer.
• For a square-free positive integer d, the ring of integers of ${\displaystyle \mathbb {Q} [{\sqrt {-d}}]}$ will fail to be a UFD unless d is a Heegner number.
• The ring of formal power series over the complex numbers is a UFD, but the subring of those that converge everywhere, in other words the ring of entire functions in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be finite, e.g.:

  ${\displaystyle \sin \pi z=\pi z\prod _{n=1}^{\infty }\left(1-{{z^{2}} \over {n^{2}}}\right).}$

Properties

Some concepts defined for integers can be generalized to UFDs:

• In UFDs, every irreducible element is prime. (In any integral domain, every prime element is irreducible, but the converse does not always hold. For instance, the element z ∈ K[x, y, z]/(z² − xy) is irreducible, but not prime.) Note that this has a partial converse: a domain satisfying the ACCP is a UFD if and only if every irreducible element is prime.
• Any two elements of a UFD have a
  associated
  .
• Any UFD is
  coefficients
  in R, then k is an element of R.
• Let S be a
  localization
  S⁻¹A is a UFD. A partial converse to this also holds; see below.

Equivalent conditions for a ring to be a UFD

A

1. A is a UFD.
2. Every nonzero prime ideal of A contains a prime element.^[5]
3. A satisfies
   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 satisfying ACCP.
7. A is a Schreier domain,^[6] 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
    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.^[7]

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 every prime ideal is generated by a prime element in a PID.

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 a height one prime ideal (induction on height) that is principal. By (2), the ring is a UFD.

See also

• Parafactorial local ring
• Noncommutative unique factorization domain

Citations

References