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{{Short description|Type of integral ___domain}}

In [[mathematics]], a '''unique factorization ___domain (UFD)''' is, roughly speaking, a [[commutative ring]] in which every element, with special exceptions, can be uniquely written as a product of [[prime element]]s, analogous to the [[fundamental theorem of arithmetic]] for the [[integer]]s. UFDs are sometimes called '''factorial rings''', following the terminology of [[Nicolas Bourbaki|Bourbaki]].

{{Redirect|Unique factorization|the uniqueness of integer factorization|fundamental theorem of arithmetic}}

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{{Algebraic structures |Ring}}

In [[mathematics]], a '''unique factorization ___domain''' ('''UFD''') (also sometimes called a '''factorial ring''' following the terminology of [[Nicolas Bourbaki|Bourbaki]]) is a [[Ring (mathematics)|ring]] in which a statement analogous to the [[fundamental theorem of arithmetic]] holds. Specifically, a UFD is an [[integral ___domain]] (a [[zero ring|nontrivial]] [[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]]s, uniquely up to order and units.

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

A unique factorization ___domain is a specific type of integral ___domain, and can be characterized by the following (not necessarily exhaustive) chain of [[subclass (set theory)|class inclusions]]:

Unique factorization domains appear in the following chain of [[subclass (set theory)|class inclusions]]:

* '''[[integral ___domain]]s''' ⊃ '''[[unique factorization ___domain]]s''' ⊃ '''[[principal ideal ___domain]]s''' ⊃ '''[[Euclidean ___domain]]s''' ⊃ '''[[field (mathematics)|field]]s'''

{{Commutative ring classes}}

== Definition ==

Formally, a unique factorization ___domain is defined to be an [[integral ___domain]] ''R'' in which every non-zero non-[[unit (ring theory)|unit]]element ''x'' of ''R'' which is not a unit can be written as a finite product of [[irreducible element]]s ''p''_''i'' of ''R'':

: ''x'' = ''p''₁ ''p''₂ ...⋅⋅⋅ ''p''_''n'' with {{nowrap|''n'' ≥ 1}}

and this representation is unique in the following sense: if
If ''q''₁, ..., ''q''_''m'' are irreducible elements of ''R'' such that

: ''x'' = ''q''₁ ''q''₂ ⋅⋅⋅ ''q''_''m'' with {{nowrap|''m'' ≥ 1}},

then {{nowrap|1=''m'' = ''n''}}, and there exists a [[bijective|bijective map]] {{nowrap|''φ'' : {{mset|1, ..., ''n''}} → {{mset|1, ..., ''m''}}}} such that ''p''_''i'' is [[Unit_(ring_theory)#Associatedness|associated]] to ''q''_{''φ''(''i'')} for {{nowrap|''i'' ∈ {{mset|1, ..., ''n''}}}}.

:''x'' = ''q''₁ ''q''₂ ... ''q''_''m'',

then ''m'' = ''n'' and there exists a [[bijective|bijective map]] &phi; : {1,...,''n''} <tt>-></tt> {1,...,''n''} such that ''p''_''i'' is [[associated element|associated]] to ''q''_&phi;(''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 UFDs:

* All [[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]] ring {{nowrap|''K''{{brackets|''X''₁, ..., ''X''_''n''}}}} 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''² + ''y''³ + ''z''⁷)}} 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]] 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}}.

* 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''² + ''y''³ + ''z''⁵)}} 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''² + ''y''³ + ''z''⁷)}} 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 other than 2. Klein and Nagata showed that the ring {{nowrap|''R''[''X''₁, ..., ''X''_''n'']/''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''² + 2''y''² + 1)}} is a UFD, but the ring {{nowrap|''Q''(''i'')[''x'', ''y'']/(''x''² + 2''y''² + 1)}} is not. On the other hand, The ring {{nowrap|''Q''[''x'', ''y'']/(''x''² + ''y''² − 1)}} is not a UFD, but the ring {{nowrap|''Q''(''i'')[''x'', ''y'']/(''x''² + ''y''² − 1)}} is.{{sfnp|Samuel|1964|p=35|ps=}} Similarly the [[coordinate ring]] {{nowrap|'''R'''[''X'', ''Y'', ''Z'']/(''X''² + ''Y''² + ''Z''² − 1)}} of the 2-dimensional [[sphere|real sphere]] is a UFD, but the coordinate ring {{nowrap|'''C'''[''X'', ''Y'', ''Z'']/(''X''² + ''Y''² + ''Z''² − 1)}} of the complex sphere is not.

* Suppose that the variables ''X''_''i'' are given weights ''w''_''i'', and {{nowrap|''F''(''X''₁, ..., ''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 module|finitely generated]] [[projective module]] over ''R'' is [[free module|free]] or ''c'' is 1 mod ''w'', the ring {{nowrap|''R''[''X''₁, ..., ''X''_''n'', ''Z'']/(''Z''^''c'' − ''F''(''X''₁, ..., ''X''_''n''))}} is a UFD.{{sfnp|Samuel|1964|p=31|ps=}}

=== Non-examples ===

Most rings familiar from elementary mathematics are UFDs:

* The [[quadratic integer ring]] <math>\mathbb Z[\sqrt{-5}]</math> of all [[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]] of <math> \mathbb Q[\sqrt{-d}]</math> will fail to be a UFD unless ''d'' is a [[Heegner number]].

* All [[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.

* 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 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.:

* Any [[field (mathematics)|field]] is trivially a UFD, since every non-zero element is a unit. Examples of fields include [[rational number]]s, [[real number]]s, and [[complex number]]s.

*: <math>\sin \pi z = \pi z \prod_{n=1}^{\infty} \left(1-{{z^2}\over{n^2}}\right).</math>

* If ''R'' is a UFD, then so is ''R''[''x''], the [[Polynomial ring|ring of polynomials]] with coefficients in ''R''. A special case of this, due to the above, is that the polynomial ring over any field is a UFD.

Further examples of UFDs are:

* The [[formal power series]] ring ''K''<nowiki>[[</nowiki>''X''₁,...,''X''_''n''<nowiki>]]</nowiki> over a field ''K''.

* The ring of functions in a fixed number of complex variables [[holomorphic function|holomorphic]] at the origin is a UFD.

*By [[mathematical induction|induction]] one can show that the polynomial rings '''Z'''[''X''₁, ..., ''X''_''n''] as well as ''K''[''X''₁, ..., ''X''_''n''] (''K'' a field) are UFDs. (Any polynomial ring with more than one variable is an example of a UFD that is not a principal ideal ___domain.)

== Counterexamples ==

*The ring <math>\mathbb Z[\sqrt{-5}]</math> of all complex numbers of the form <math>a+ib\sqrt{5}</math>, where ''a'' and ''b'' are integers. Then 6 factors as both (2)(3) and as <math>\left(1+i\sqrt{5}\right)\left(1-i\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+i\sqrt{5}</math>, and <math>1-i\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. See also [[algebraic integer]].

*Most [[ideal (ring theory)|factor ring]]s of a polynomial ring are not UFDs. Here is an example:

:Let <math>R</math> be any commutative ring. Then <math>R[X,Y,Z,W]/(XY-ZW)</math> is not a UFD. The proof is in two parts.

:First we must show <math>X</math>, <math>Y</math>, <math>Z</math>, and <math>W</math> are all irreducible. Grade <math>R[X,Y,Z,W]/(XY-ZW)</math> by degree. Assume for a contradiction that <math>X</math> has a factorization into two non-zero non-units. Since it is degree one, the two factors must be a degree one element <math>\alpha X + \beta Y + \gamma Z + \delta W</math> and a degree zero element <math>r</math>. This gives <math>X = r\alpha X + r\beta Y + r\gamma Z + r\delta W</math>. In <math>R[X,Y,Z,W]</math>, then, the degree one element <math>(r\alpha-1) X + r\beta Y + r\gamma Z + r\delta W</math> must be an element of the ideal <math>(XY-ZW)</math>, but the non-zero elements of that ideal are degree two and higher. Consequently, <math>(r\alpha-1) X + r\beta Y + r\gamma Z + r\delta W</math> must be zero in <math>R[X,Y,Z,W]</math>. That implies that <math>r\alpha = 1</math>, so <math>r</math> is a unit, which is a contradiction. <math>Y</math>, <math>Z</math>, and <math>W</math> are irreducible by the same argument.

:Next, the element <math>XY</math> equals the element <math>ZW</math> because of the relation <math>XY - ZW = 0</math>. That means that <math>XY</math> and <math>ZW</math> are two different factorizations of the same element into irreducibles, so <math>R[X,Y,Z,W]/(XY-ZW)</math> is not a UFD.

*The ring of holomorphic functions in a single complex variable is not a UFD, since there exist holomorphic functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be finite, i.e: <math>\sin \pi z = \pi z \prod_{n=1}^{\infty} (1-{{z^2}\over{n^2}})</math>

== Properties ==

Some concepts defined for integers can be generalized to UFDs:

* In UFDs, every [[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''² − ''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]] and a [[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 element|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 [[Polynomial root|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 of a ring|localization]] ''S''⁻¹''A'' is a UFD. A partial converse to this also holds; see below.

== Equivalent conditions for a ring to be a UFD ==

* In UFDs, every [[irreducible element]] is [[prime element|prime]]. (In any integral ___domain, every prime element is irreducible, but the converse does not always hold.) Note that this has a partial converse: any [[Noetherian ring | Noetherian ___domain]] is a UFD iff every irreducible element is prime (this is one proof of the implication PID <math>\Rightarrow</math> UFD).

A [[Noetherian ring|Noetherian]] integral ___domain is a UFD if and only if every [[height (ring theory)|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:

* 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 [[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.

# ''A'' is a UFD.

# Every nonzero [[prime ideal]] of ''A'' contains a [[prime element]].{{refn|[[Irving Kaplansky|Kaplansky]]}}

# ''A'' satisfies [[ascending chain condition on principal ideals]] (ACCP), and the [[localization of a ring|localization]] ''S''^&minus;1''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''₁ ''x''₂}} so that ''x''₁ divides ''y'' and ''x''₂ 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]] 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.

* Any UFD is [[integrally closed]]. In other words, if R is an [[integral ___domain]] with [[quotient field]] K, and if an element k in K is a [[root#Mathematics|root]] of a [[monic#Mathematics|monic polynomial]] with [[coefficients]] in R, then k is an element of R.

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.

== Equivalent conditions for a ring to be a UFD ==

== See also ==

Under some circumstances, it is possible to give equivalent conditions for a ring to be a UFD.

* [[Parafactorial local ring]]

* [[Noncommutative unique factorization ___domain]]

== Citations ==

* A [[Noetherian]] integral ___domain is a UFD [[if and only if]] every [[height (ring theory)|height]] 1 [[prime ideal]] is principal.

{{reflist}}

== References ==

* 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.

{{refbegin}}

* {{cite book | last1=Artin | first1=Michael |authorlink = Michael Artin| 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. | author-link=Harold Edwards (mathematician)| title=Divisor Theory | publisher=Birkhäuser | publication-place=Boston | date=1990 | isbn=978-0-8176-3448-3}}

* {{cite book | last1=Hartley | first1=B. | author-link=Brian Hartley | 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 | author1-link=Pierre Samuel | 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 | author1-link=Pierre Samuel | 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}}

{{Authority control}}

* There is a nice [[two-sided ideal|ideal]]-theoretic characterization of UFDs, due to [[Kaplansky]]. If R is an [[integral ___domain]], then R is a UFD if and only if every nonzero [[prime ideal]] of R has a nonzero [[prime element]].

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