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* 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>\textstyle \mathbb{Z}\
* 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.
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