Editing Unique factorization domain (section)

== 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 &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]].
* 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>