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Unique factorization domain
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{{Short description|Type of integral domain}} {{Redirect|Unique factorization|the uniqueness of integer factorization|fundamental theorem of arithmetic}} {{pp-sock|small=yes}} {{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]]. 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]] ''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]]s ''p''<sub>''i''</sub> of ''R'': : ''x'' = ''p''<sub>1</sub> ''p''<sub>2</sub> ⋅⋅⋅ ''p''<sub>''n''</sub> with {{nowrap|''n'' ≥ 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'' ≥ 1}}, then {{nowrap|1=''m'' = ''n''}}, and there exists a [[bijective|bijective map]] {{nowrap|''φ'' : {{mset|1, ..., ''n''}} → {{mset|1, ..., ''m''}}}} such that ''p''<sub>''i''</sub> is [[Unit_(ring_theory)#Associatedness|associated]] to ''q''<sub>''φ''(''i'')</sub> for {{nowrap|''i'' ∈ {{mset|1, ..., ''n''}}}}. == 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''<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]] is a UFD. * <math>\mathbb{Z}\left[e^{\frac{2 \pi i}{n}}\right]</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''<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 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]] 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''<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]]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 −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]]. * 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.: *: <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]] 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'' ∈ ''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]] 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''<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 (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: # ''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''<sup>−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]] 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 |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}} [[Category:Ring theory]] [[Category:Algebraic number theory]] [[Category:factorization]]
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