Unique factorization domain

Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R which is not a unit can be written as a finite product of irreducible elements p_i of R:

x = p₁ p₂ ⋅⋅⋅ p_n with n ≥ 1

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 with m ≥ 1,

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}.

Most rings familiar from elementary mathematics are UFDs:

• All principal ideal domains, hence all Euclidean domains, are UFDs. In particular, the 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.
• ${\displaystyle \textstyle \mathbb {Z} {\bigl [}e^{2\pi i/n}{\bigr ]}}$ is a UFD for all integers 1 ≤ n ≤ 22, but not for n = 23.
• Mori showed that if the completion of a Zariski ring, such as a Noetherian local ring, is a UFD, then the ring is a UFD.^[1] The converse of this is not true: there are Noetherian local rings that are UFDs but whose completions are not. The question of when this happens is rather subtle: for example, for the 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.^[2] Similarly the 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]

• 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.:

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

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 greatest common divisor and a least common multiple. Here, a greatest common divisor of a and b is an element d that 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.
• Any UFD is integrally closed. In other words, if R is a UFD with quotient field K, and if an element k in K is a root of a monic polynomial with coefficients in R, then k is an element of R.
• Let S be a multiplicatively closed subset of a UFD A. Then the localization S⁻¹A is a UFD. A partial converse to this also holds; see below.

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

In general, for an integral domain A, the following conditions are equivalent:

1. A is a UFD.
2. Every nonzero prime ideal of A contains a prime element.^[5]
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 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 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.^[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.

• Parafactorial local ring
• Noncommutative unique factorization domain