Unique factorization ___domain: Difference between revisions

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== 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 a [[Unit (ring theory)|unit]] ''u'' and zero or more [[irreducible element]]s ''p''_''i'' of ''R'':

: ''x'' = ''u'' ''p''₁ ''p''₂ ⋅⋅⋅ ''p''_''n'' with {{nowrap|''n'' ≥ 01}}

and this representation is unique in the following sense:

If ''q''₁, ..., ''q''_''m'' are irreducible elements of ''R'' and ''w'' is a unit such that

: ''x'' = ''w'' ''q''₁ ''q''₂ ⋅⋅⋅ ''q''_''m'' with {{nowrap|''m'' ≥ 01}},

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''}}}}.

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* 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}\leftbigl[e^{\frac{2 \pi i}{/n}}\rightbigr]</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 ===