|
|
| Line 1: |
Line 1: |
| In [[mathematics]], a '''unique factorization domain (UFD)''' is a [[commutative ring]] in which every non-zero non-unit element can be written as a product of [[prime element]]s (or [[irreducible element]]s), uniquely up to order and units, analogous to the [[fundamental theorem of arithmetic]] for the [[integer]]s. UFDs are sometimes called '''factorial rings''', following the terminology of [[Nicolas Bourbaki|Bourbaki]].
| | My name is Soon and I am studying Dance and Journalism at Montigny-Le-Bretonneux / France.<br><br>Feel free to visit my web-site: [http://ebontalifarro.livejournal.com/ Ebon Talifarro] |
| | |
| Unique factorization domains appear in the following chain of [[subclass (set theory)|class inclusions]]:
| |
| | |
| : '''[[Commutative ring]]s''' ⊃ '''[[integral domain]]s''' ⊃ '''[[integrally closed domain]]s''' ⊃ '''unique factorization domains''' ⊃ '''[[principal ideal domain]]s''' ⊃ '''[[Euclidean domain]]s''' ⊃ '''[[field (mathematics)|field]]s'''
| |
| | |
| == Definition ==
| |
| Formally, a unique factorization domain is defined to be an [[integral domain]] ''R'' in which every non-zero ''x'' of ''R'' can be written as a product (an [[empty product]] if ''x'' is a unit) of [[irreducible element]]s ''p''<sub>i</sub> of ''R'' and a [[Unit_(ring_theory)|unit]] ''u'':
| |
| :''x'' = ''u'' ''p''<sub>1</sub> ''p''<sub>2</sub> ... ''p''<sub>''n''</sub> with ''n''≥0
| |
| and this representation is unique in the following sense:
| |
| If ''q''<sub>1</sub>,...,''q''<sub>''m''</sub> are irreducible elements of ''R'' and ''w'' is a unit such that
| |
| | |
| :''x'' = ''w'' ''q''<sub>1</sub> ''q''<sub>2</sub> ... ''q''<sub>''m''</sub> with ''m''≥0,
| |
| | |
| then ''m'' = ''n'', ''u'' = ''w'' and there exists a [[bijective|bijective map]] φ : {1,...,''n''} <tt>-></tt> {1,...,''m''} such that ''p''<sub>''i''</sub> is [[associated element|associated]] to ''q''<sub>φ(''i'')</sub> for ''i'' ∈ {1, ..., ''n''}.
| |
| | |
| The uniqueness part is usually 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 element can be written as a product of a unit and [[prime element]]s 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 iteration, a polynomial ring in any number of variables over any UFD (and in particular over a field) is a UFD.
| |
| * The [[Auslander–Buchsbaum theorem]] states that every [[regular local ring]] is a UFD.
| |
| * The [[formal power series]] ring ''K''<nowiki>[[</nowiki>''X''<sub>1</sub>,...,''X''<sub>''n''</sub><nowiki>]]</nowiki> 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'']/(''x''<sup>2</sup>+''y''<sup>3</sup>+''z''<sup>7</sup>) at the [[prime ideal]] (''x'',''y'',''z'') then ''R'' is a local ring that is a UFD, but the formal power series ring ''R''<nowiki>[[</nowiki>''X''<nowiki>]]</nowiki> over ''R'' is not a UFD.
| |
| *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.<ref>Bourbaki, 7.3, no 6, Proposition 4.</ref> 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 ''k''[''x'',''y'',''z'']/(''x''<sup>2</sup>+''y''<sup>3</sup>+''z''<sup>5</sup>) 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'']/(''x''<sup>2</sup>+''y''<sup>3</sup>+''z''<sup>7</sup>) at the prime ideal (''x'',''y'',''z'') the local ring is a UFD but its completion is not.
| |
| *A non-example: the [[quadratic integer ring]] <math>\mathbb Z[\sqrt{-5}]</math> of all complex numbers 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.<ref>{{cite book|last=Artin|first=Michael|title=Algebra|date=2011|publisher=Prentice Hall|isbn=978-0-13-241377-0|page=360}}</ref> See also [[algebraic integer]].
| |
| *Let <math>R</math> be any field of characteristic not 2. Klein and Nagata showed that the ring ''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 ''n''=4 the ring need not be a UFD. For example, <math>R[X,Y,Z,W]/(XY-ZW)</math> is not a UFD, because the element <math>XY</math> equals the element <math>ZW</math> so that <math>XY</math> and <math>ZW</math> are two different factorizations of the same element into irreducibles.
| |
| *The ring of formal power series over the complex numbers is factorial, but the [[subring]] of those that converge everywhere, in other words the ring of [[holomorphic function]]s 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, e.g.:
| |
| *:<math>\sin \pi z = \pi z \prod_{n=1}^{\infty} \left(1-{{z^2}\over{n^2}}\right)</math>.
| |
| *The ring ''Q''[''x'',''y'']/(''x''<sup>2</sup>+2''y''<sup>2</sup>+1) is factorial, but the ring ''Q''(''i'')[''x'',''y'']/(''x''<sup>2</sup>+2''y''<sup>2</sup>+1) is not. On the other hand, The ring ''Q''[''x'',''y'']/(''x''<sup>2</sup>+''y''<sup>2</sup>–1) is not factorial, but the ring ''Q''(''i'')[''x'',''y'']/(''x''<sup>2</sup>+''y''<sup>2</sup>–1) is {{harv|Samuel|1964|loc=p.35}}. Similarly the [[coordinate ring]] '''R'''[''X'',''Y'',''Z'']/(''X''<sup>2</sup>+''Y''<sup>2</sup>+''Z''<sup>2</sup>–1) of the 2-dimensinal real sphere is factorial, but the coordinate ring '''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 ''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 [[projective module]] over ''R'' is free or ''c'' is 1 mod ''w'', the ring ''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 factorial ring {{harv|Samuel|1964|loc=p.31}}.
| |
| | |
| == 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 <math>z\in K[x,y,z]/(z^2-xy)</math> is irreducible, but not prime.) Note that this has a partial converse: a [[Noetherian ring | Noetherian domain]] is a UFD if every irreducible element is prime.
| |
| * 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 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 [[root#Mathematics|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]] <math>S^{-1}A</math> 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 below). 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]].
| |
| | |
| There are also equivalent conditions for non-noetherian integral domains. Let ''A'' be an integral domain. Then the following are equivalent.
| |
| # ''A'' is a UFD.
| |
| # Every nonzero [[prime ideal]] of ''A'' contains a [[prime element]]. ([[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 (ACCP) and every [[irreducible element|irreducible]] is [[prime element|prime]].
| |
| # ''A'' is a [[GCD domain]] (i.e., any two elements have a greatest common divisor) satisfying (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 ''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 every nonzero nonunit can be expressed as a finite product of irreducible elements (that is, ''A'' is [[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.<ref>Bourbaki, 7.3, no 2, Theorem 1.</ref>
| |
| | |
| 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, in a PID, every prime ideal is generated by a prime element.
| |
| | |
| 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 height one prime ideal (induction on height), which is principal. By (2), the ring is a UFD.
| |
| | |
| ==See also==
| |
| | |
| *[[Parafactorial local ring]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| * {{cite book | author=N. Bourbaki |title=Commutative algebra}}
| |
| * {{cite book | author=B. Hartley | authorlink=Brian Hartley | coauthors=T.O. Hawkes | title=Rings, modules and linear algebra | publisher=Chapman and Hall | year=1970 | isbn=0-412-09810-5 }} Chap. 4.
| |
| * Chapter II.5 of {{Lang Algebra|edition=3}}
| |
| * {{cite book | author=David Sharpe | title=Rings and factorization | 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 | id={{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 | issue=75 | pages=945–952 | postscript=<!--None-->}}
| |
| | |
| [[Category:Abstract algebra]]
| |
| [[Category:Ring theory]]
| |
| [[Category:Number theory]]
| |