Inmathematics, aunique factorization domain (UFD) (also sometimes called afactorial ring following the terminology ofBourbaki) is aring in which a statement analogous to thefundamental theorem of arithmetic holds. Specifically, a UFD is anintegral domain (anontrivialcommutative 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 ofirreducible elements, uniquely up to order and units.

Important examples of UFDs are the integers andpolynomial rings in one or more variables with coefficients coming from the integers or from afield.

Unique factorization domains appear in the following chain ofclass inclusions:

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

Formally, a unique factorization domain is defined to be anintegral domainR in which every non-zero elementx ofR which is not a unit can be written as a finite product ofirreducible elementsp_i ofR:

x =p₁p₂ ⋅⋅⋅p_n withn ≥ 1

and this representation is unique in the following sense:Ifq₁, ...,q_m are irreducible elements ofR such that

x =q₁q₂ ⋅⋅⋅q_m withm ≥ 1,

thenm =n, and there exists abijective mapφ : {1, ...,n} → {1, ...,m} such thatp_i isassociated toq_φ(i) fori ∈ {1, ...,n}.

Most rings familiar from elementary mathematics are UFDs:

• Allprincipal ideal domains, hence allEuclidean domains, are UFDs. In particular, theintegers (also seeFundamental theorem of arithmetic), theGaussian integers and theEisenstein integers are UFDs.
• IfR is a UFD, then so isR[X], thering of polynomials with coefficients inR. UnlessR 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.
• Theformal power series ringK[[X₁, ...,X_n]] over a fieldK (or more generally over aregular 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 islocal. For example, ifR is the localization ofk[x,y,z]/(x² +y³ +z⁷) at theprime ideal(x,y,z) thenR is a local ring that is a UFD, but the formal power series ringR[[X]] overR is not a UFD.
• TheAuslander–Buchsbaum theorem states that everyregular local ring is a UFD.
• ${\displaystyle \mathbb{Z}\left[e^{\frac{2\pi i}{n}}\right]}$ is a UFD for all integers1 ≤n ≤ 22, but not forn = 23.
• Mori showed that if the completion of aZariski ring, such as aNoetherian 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 thelocalization ofk[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 ofk[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 ringR[X₁, ...,X_n]/Q is a UFD wheneverQ is a nonsingular quadratic form in theXs andn is at least 5. Whenn = 4, the ring need not be a UFD. For example,R[X,Y,Z,W]/(XY −ZW) is not a UFD, because the elementXY equals the elementZW so thatXY andZW are two different factorizations of the same element into irreducibles.
• The ringQ[x,y]/(x² + 2y² + 1) is a UFD, but the ringQ(i)[x,y]/(x² + 2y² + 1) is not. On the other hand, The ringQ[x,y]/(x² +y² − 1) is not a UFD, but the ringQ(i)[x,y]/(x² +y² − 1) is.^[2] Similarly thecoordinate ringR[X,Y,Z]/(X² +Y² +Z² − 1) of the 2-dimensionalreal sphere is a UFD, but the coordinate ringC[X,Y,Z]/(X² +Y² +Z² − 1) of the complex sphere is not.
• Suppose that the variablesX_i are given weightsw_i, andF(X₁, ...,X_n) is ahomogeneous polynomial of weightw. Then ifc is coprime tow andR is a UFD and either everyfinitely generatedprojective module overR isfree orc is 1 modw, the ringR[X₁, ...,X_n,Z]/(Z^c −F(X₁, ...,X_n)) is a UFD.^[3]

• Thequadratic integer ring${\displaystyle \mathbb{Z}[\sqrt{-5}]}$ of allcomplex numbers of the form${\displaystyle a+b\sqrt{-5}}$, wherea andb 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${\displaystyle 1-\sqrt{-5}}$ areassociate. It is not hard to show that all four factors are irreducible as well, though this may not be obvious.^[4] See alsoAlgebraic integer.
• For asquare-free positive integerd, thering of integers of${\displaystyle \mathbb{Q}[\sqrt{-d}]}$ will fail to be a UFD unlessd is aHeegner number.
• The ring of formal power series over the complex numbers is a UFD, but thesubring of those that converge everywhere, in other words the ring ofentire 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.:

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

Some concepts defined for integers can be generalized to UFDs:

• In UFDs, everyirreducible element isprime. (In any integral domain, every prime element is irreducible, but the converse does not always hold. For instance, the elementz ∈K[x,y,z]/(z² −xy) is irreducible, but not prime.) Note that this has a partial converse: a domain satisfying theACCP is a UFD if and only if every irreducible element is prime.
• Any two elements of a UFD have agreatest common divisor and aleast common multiple. Here, a greatest common divisor ofa andb is an elementd thatdivides botha andb, and such that every other common divisor ofa andb dividesd. All greatest common divisors ofa andb areassociated.
• Any UFD isintegrally closed. In other words, ifR is a UFD withquotient fieldK, and if an elementk inK is aroot of amonic polynomial withcoefficients inR, thenk is an element ofR.
• LetS be amultiplicatively closed subset of a UFDA. Then thelocalizationS⁻¹A is a UFD. A partial converse to this also holds; see below.

ANoetherian integral domain is a UFD if and only if everyheight 1prime ideal is principal (a proof is given at the end). Also, aDedekind domain is a UFD if and only if itsideal class group is trivial. In this case, it is in fact aprincipal ideal domain.

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

1. A is a UFD.
2. Every nonzeroprime ideal ofA contains aprime element.^[5]
3. A satisfiesascending chain condition on principal ideals (ACCP), and thelocalizationS⁻¹A is a UFD, whereS is amultiplicatively closed subset ofA generated by prime elements. (Nagata criterion)
4. A satisfiesACCP and everyirreducible isprime.
5. A isatomic and everyirreducible isprime.
6. A is aGCD domain satisfyingACCP.
7. A is aSchreier domain,^[6] andatomic.
8. A is apre-Schreier domain andatomic.
9. A has adivisor theory in which every divisor is principal.
10. A is aKrull domain in which everydivisorial 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

• Ring theory
• Algebraic number theory
• Factorization

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