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{{Short description|Type of integral domain}}
{{Redirect|Unique factorization|the uniqueness of integer factorization|fundamental theorem of arithmetic}}
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{{Algebraic structures |Ring}}
In [[Mathematics|mathematics]], a '''unique factorization domain''' ('''UFD''') (also sometimes called a '''factorial ring''' following the terminology of Bourbaki) is a [[Ring (mathematics)|ring]] in which a statement analogous to the [[Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] holds. Specifically, a UFD is an [[Integral domain|integral domain]] (a [[Zero ring|nontrivial]] [[Commutative ring|commutative ring]] in which the product of any two non-zero elements is non-zero) in which every non-zero non-[[Unit (ring theory)|unit]] element can be written as a product of [[Irreducible element|irreducible element]]s, uniquely up to order and units.

Important examples of UFDs are the integers and [[Polynomial ring|polynomial ring]]s in one or more variables with coefficients coming from the integers or from a [[Field (mathematics)|field]].

Unique factorization domains appear in the following chain of [[Subclass (set theory)|class inclusions]]:
{{Commutative ring classes}}

== Definition ==
Formally, a unique factorization domain is defined to be an [[Integral domain|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 element|irreducible element]]s ''p''<sub>''i''</sub> of ''R'':
: ''x'' = ''p''<sub>1</sub> ''p''<sub>2</sub> ⋅⋅⋅ ''p''<sub>''n''</sub> with {{nowrap|''n'' &ge; 1}}
and this representation is unique in the following sense:
If ''q''<sub>1</sub>, ..., ''q''<sub>''m''</sub> are irreducible elements of ''R'' such that
: ''x'' = ''q''<sub>1</sub> ''q''<sub>2</sub> ⋅⋅⋅ ''q''<sub>''m''</sub> with {{nowrap|''m'' &ge; 1}},
then {{nowrap|1=''m'' = ''n''}}, and there exists a bijective map {{nowrap|''&phi;'' : {{mset|1, ..., ''n''}} → {{mset|1, ..., ''m''}}}} such that ''p''<sub>''i''</sub> is [[Unit (ring theory)#Associatedness|associated]] to ''q''<sub>''&phi;''(''i'')</sub> for {{nowrap|''i'' &isin; {{mset|1, ..., ''n''}}}}.

== Examples ==
Most rings familiar from elementary mathematics are UFDs:
* All [[Principal ideal domain|principal ideal domain]]s, hence all [[Euclidean domain]]s, are UFDs. In particular, the integers (also see ''[[Fundamental theorem of arithmetic]]''), the [[Gaussian integer]]s and the [[Eisenstein integer]]s are UFDs.
* If ''R'' is a UFD, then so is ''R''[''X''], the [[Polynomial ring|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|formal power series]] ring {{nowrap|''K''{{brackets|''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>}}}} over a field ''K'' (or more generally over a [[Regular local ring#Regular ring|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 ring|local]]. For example, if ''R'' is the localization of {{nowrap|''k''[''x'', ''y'', ''z'']/(''x''<sup>2</sup> + ''y''<sup>3</sup> + ''z''<sup>7</sup>)}} at the prime ideal {{nowrap|(''x'', ''y'', ''z'')}} then ''R'' is a local ring that is a UFD, but the formal power series ring ''R''{{brackets|''X''}} over ''R'' is not a UFD.
* The [[Auslander–Buchsbaum theorem]] states that every [[Regular local ring|regular local ring]] is a UFD.
* <math>\textstyle \mathbb{Z}\bigl[e^{2 \pi i/n}\bigr]</math> is a UFD for all integers {{nowrap|1 ≤ ''n'' ≤ 22}}, but not for {{nowrap|1=''n'' = 23}}. This is the [[Ring of integers|ring of integers]] of the [[Cyclotomic field|cyclotomic field]] <math>\mathbb{Q}(e^{2 \pi i/n})</math>. For rings of integers of other [[Number field|number field]]s, see ''List of number fields with class number one''.
* Mori showed that if the completion of a [[Zariski ring]], such as a [[Noetherian ring|Noetherian local ring]], is a UFD, then the ring is a UFD.{{sfnp|Bourbaki|1972|loc=7.3, no 6, Proposition 4|ps=}} 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 a ring|localization]] of {{nowrap|''k''[''x'', ''y'', ''z'']/(''x''<sup>2</sup> + ''y''<sup>3</sup> + ''z''<sup>5</sup>)}} at the prime ideal {{nowrap|(''x'', ''y'', ''z'')}}, both the local ring and its completion are UFDs, but in the apparently similar example of  the localization of {{nowrap|''k''[''x'', ''y'', ''z'']/(''x''<sup>2</sup> + ''y''<sup>3</sup> + ''z''<sup>7</sup>)}} at the prime ideal {{nowrap|(''x'', ''y'', ''z'')}} the local ring is a UFD but its completion is not.
* Let <math>R</math> be a field of any [[Characteristic (algebra)|characteristic]] other than 2. Klein and Nagata showed that the ring {{nowrap|''R''[''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>]/''Q''}} is a UFD whenever ''Q'' is a nonsingular quadratic form in the ''X''s and ''n'' is at least 5. When {{nowrap|1=''n'' = 4}}, the ring need not be a UFD. For example, {{nowrap|''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 {{nowrap|''Q''[''x'', ''y'']/(''x''<sup>2</sup> + 2''y''<sup>2</sup> + 1)}} is a UFD, but the ring {{nowrap|''Q''(''i'')[''x'', ''y'']/(''x''<sup>2</sup> + 2''y''<sup>2</sup> + 1)}} is not. On the other hand, The ring {{nowrap|''Q''[''x'', ''y'']/(''x''<sup>2</sup> + ''y''<sup>2</sup> − 1)}} is not a UFD, but the ring {{nowrap|''Q''(''i'')[''x'', ''y'']/(''x''<sup>2</sup> + ''y''<sup>2</sup> − 1)}} is.{{sfnp|Samuel|1964|p=35|ps=}} Similarly the coordinate ring {{nowrap|'''R'''[''X'', ''Y'', ''Z'']/(''X''<sup>2</sup> + ''Y''<sup>2</sup> + ''Z''<sup>2</sup> − 1)}} of the 2-dimensional [[Sphere|real sphere]] is a UFD, but the coordinate ring {{nowrap|'''C'''[''X'', ''Y'', ''Z'']/(''X''<sup>2</sup> + ''Y''<sup>2</sup> + ''Z''<sup>2</sup> − 1)}} of the complex sphere is not.
* Suppose that the variables ''X''<sub>''i''</sub> are given weights ''w''<sub>''i''</sub>, and {{nowrap|''F''(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}} is a [[Homogeneous polynomial|homogeneous polynomial]] of weight ''w''. Then if ''c'' is coprime to ''w'' and ''R'' is a UFD and either every [[Finitely generated module|finitely generated]] [[Projective module|projective module]] over ''R'' is [[Free module|free]] or ''c'' is 1 mod ''w'', the ring {{nowrap|''R''[''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>, ''Z'']/(''Z''<sup>''c''</sup> − ''F''(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>))}} is a UFD.{{sfnp|Samuel|1964|p=31|ps=}}

=== Non-examples ===
* The quadratic integer ring <math>\mathbb Z[\sqrt{-5}]</math> of all [[Complex number|complex number]]s of the form <math>a+b\sqrt{-5}</math>, where ''a'' and ''b'' are integers, is not a UFD because 6 factors as both 2×3 and as <math>\left(1+\sqrt{-5}\right)\left(1-\sqrt{-5}\right)</math>. These truly are different factorizations, because the only units in this ring are 1 and &minus;1; thus, none of 2, 3, <math>1+\sqrt{-5}</math>, and <math>1-\sqrt{-5}</math> are [[Unit (ring theory)|associate]]. It is not hard to show that all four factors are irreducible as well, though this may not be obvious.{{sfnp|Artin|2011|p=360|ps=}} See also ''[[Algebraic integer]]''.
* For a [[Square-free integer|square-free positive integer]] ''d'', the [[Ring of integers|ring of integers]] of <math> \mathbb Q[\sqrt{-d}]</math> will fail to be a UFD unless ''d'' is one of the nine [[Heegner number]]s.
* The ring of formal power series over the complex numbers is a UFD, but the [[Subring|subring]] of those that converge everywhere, in other words the ring of [[Entire function|entire function]]s 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.:
*: <math>\sin \pi z = \pi z \prod_{n=1}^{\infty} \left(1-{{z^2}\over{n^2}}\right).</math>

== Properties ==
Some concepts defined for integers can be generalized to UFDs:
* In UFDs, every [[Irreducible element|irreducible element]] is [[Prime element|prime]]. (In any integral domain, every prime element is irreducible, but the converse does not always hold.  For instance, the element {{nowrap|''z'' &isin; ''K''[''x'', ''y'', ''z'']/(''z''<sup>2</sup> − ''xy'')}} is irreducible, but not prime.) Note that this has a partial converse: a domain satisfying the [[Ascending chain condition on principal ideals|ACCP]] is a UFD if and only if every irreducible element is prime.
* Any two elements of a UFD have a [[Greatest common divisor|greatest common divisor]] and a [[Least common multiple|least common multiple]]. Here, a greatest common divisor of ''a'' and ''b'' is an element ''d'' that [[Divisor|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 domain|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|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 of a ring|localization]] ''S''<sup>−1</sup>''A'' is a UFD. A partial converse to this also holds; see below.

== Equivalent conditions for a ring to be a UFD ==
A [[Noetherian ring|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|ideal class group]] is trivial. In this case, it is in fact a [[Principal ideal domain|principal ideal domain]].

In general, for an integral domain ''A'', the following conditions are equivalent:
# ''A'' is a UFD.
# Every nonzero prime ideal of ''A'' contains a [[Prime element|prime element]].{{refn|[[Biography:Irving Kaplansky|Kaplansky]]}}
# ''A'' satisfies [[Ascending chain condition on principal ideals|ascending chain condition on principal ideals]] (ACCP), and the [[Localization of a ring|localization]] ''S''<sup>&minus;1</sup>''A'' is a UFD, where ''S'' is a multiplicatively closed subset of ''A'' generated by prime elements. (Nagata criterion)
# ''A'' satisfies [[Ascending chain condition on principal ideals|ACCP]] and every [[Irreducible element|irreducible]] is [[Prime element|prime]].
# ''A'' is [[Atomic domain|atomic]] and every [[Irreducible element|irreducible]] is [[Prime element|prime]].
# ''A'' is a [[GCD domain]] satisfying [[Ascending chain condition on principal ideals|ACCP]].
# ''A'' is a [[Schreier domain]],<ref>A Schreier domain is an integrally closed integral domain where, whenever ''x'' divides ''yz'', ''x'' can be written as {{nowrap|1=''x'' = ''x''<sub>1</sub> ''x''<sub>2</sub>}} so that ''x''<sub>1</sub> divides ''y'' and ''x''<sub>2</sub> divides ''z''. In particular, a GCD domain is a Schreier domain</ref> and [[Atomic domain|atomic]].
# ''A'' is a [[Schreier domain|pre-Schreier domain]] and [[Atomic domain|atomic]].
# ''A'' has a divisor theory in which every divisor is principal.
# ''A'' is a Krull domain in which every [[Divisorial ideal|divisorial ideal]] is principal (in fact, this is the definition of UFD in Bourbaki.)
# ''A'' is a Krull domain and every prime ideal of height 1 is principal.{{sfnp|Bourbaki|1972|loc=7.3, no 2, Theorem 1}}

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 ==
{{reflist}}

== References ==
{{refbegin}}
* {{cite book | last1=Artin | first1=Michael | title=Algebra | year=2011 | publisher=Prentice Hall | isbn=978-0-13-241377-0 }}
* {{cite book | last1=Bourbaki | first1= N. | title=Commutative algebra|year=1972 |publisher=Paris, Hermann; Reading, Mass., Addison-Wesley Pub. Co |isbn=9780201006445 | url=https://archive.org/details/commutativealgeb0000bour | url-access=registration }}
* {{cite book | last=Edwards | first=Harold M. |  title=Divisor Theory | publisher=Birkhäuser | publication-place=Boston | date=1990 | isbn=978-0-8176-3448-3}}
* {{cite book | last1=Hartley | first1=B. |  author2=T.O. Hawkes  | title=Rings, modules and linear algebra | publisher=Chapman and Hall | year=1970 | isbn=0-412-09810-5 }}  Chap. 4.
* {{Lang Algebra|edition=3}} Chapter II.5
* {{cite book | last=Sharpe | first=David | title=Rings and factorization | url=https://archive.org/details/ringsfactorizati0000shar | url-access=registration | publisher=Cambridge University Press | year=1987 | isbn=0-521-33718-6 }}
* {{citation | last1=Samuel | first1=Pierre |  editor1-last=Murthy | editor1-first=M. Pavman | title=Lectures on unique factorization domains | url=http://www.math.tifr.res.in/~publ/ln/ | publisher=Tata Institute of Fundamental Research | location=Bombay | series=Tata Institute of Fundamental Research Lectures on Mathematics |mr=0214579 | year=1964 | volume=30}}
* {{cite journal | last1=Samuel | first1=Pierre |  title=Unique factorization | year=1968 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=75 | issue=9 | pages=945–952 | doi=10.2307/2315529| jstor=2315529 }}
* {{cite book | last=Weintraub | first=Steven H. | title=Factorization: Unique and Otherwise | publisher=A K Peters/CRC Press | publication-place=Wellesley, Mass. | date=2008 | isbn=978-1-56881-241-0}}
{{refend}}


[[Category:Ring theory]]
[[Category:Algebraic number theory]]

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