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There's no point in forcing the units to be part of the definition. The standard approach is to consider x to be non-zero and non-unit. |
m →Examples: lklk |
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* 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 ===
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