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For example, if R is the localization of ''k''[''x'', ''y'', ''z'']/(''x''² + ''y''³ + ''z''⁷) at the prime ideal (''x'', ''y'', ''z'') then R is a local ring that is a UFD, but the formal power series ring R over R is not a UFD.
* The Auslander-Buchsbaum theorem states that every regular local ring is a UFD.
* is a UFD for all integers 1 <= ''n'' <= 22, but not for 1=''n'' = 23. This is the ring of integers of the cyclotomic field . For rings of integers of other number fields, see List of number fields with class number one.
* Mori showed that if the completion of a Zariski ring, such as a Noetherian local ring, is a UFD, then the ring is a UFD. 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 ''k''[''x'', ''y'', ''z'']/(''x''² + ''y''³ + ''z''⁵) 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''² + ''y''³ + ''z''⁷) at the prime ideal (''x'', ''y'', ''z'') the local ring is a UFD but its completion is not.
* Let be a field of any characteristic other than 2. Klein and Nagata showed that the ring ''R''[''X''₁, ..., ''X''_''n'']/''Q'' is a UFD whenever Q is a nonsingular quadratic form in the Xs and n is at least 5. When 1=''n'' = 4, the ring need not be a UFD. For example, ''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 ''Q''[''x'', ''y'']/(''x''² + 2''y''² + 1) is a UFD, but the ring ''Q''(''i'')[''x'', ''y'']/(''x''² + 2''y''² + 1) is not. On the other hand, The ring ''Q''[''x'', ''y'']/(''x''² + ''y''² - 1) is not a UFD, but the ring ''Q''(''i'')[''x'', ''y'']/(''x''² + ''y''² - 1) is. Similarly the coordinate ring '''R'''[''X'', ''Y'', ''Z'']/(''X''² + ''Y''² + ''Z''² - 1) of the 2-dimensional real sphere is a UFD, but the coordinate ring '''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 ''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 projective module over R is free or c is 1 mod w, the ring ''R''[''X''₁, ..., ''X''_''n'', ''Z'']/(''Z''^''c'' - ''F''(''X''₁, ..., ''X''_''n'')) is a UFD.

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