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Added integral ___domain class inclusion chain for completeness and consistency (to match the other specific integral ___domain articles' class inclusion chains) |
added ring of holomorphic functions as a counterexample |
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:Next, the element <math>XY</math> equals the element <math>ZW</math> because of the relation <math>XY - ZW = 0</math>. That means that <math>XY</math> and <math>ZW</math> are two different factorizations of the same element into irreducibles, so <math>R[X,Y,Z,W]/(XY-ZW)</math> is not a UFD.
*The ring of holomorphic functions in a single complex variable is not a UFD, since there exist holomorphic functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be finite, i.e: <math>\sin \pi z = \pi z \prod_{n=1}^{\infty} (1-{{z^2}\over{n^2}})</math>
== Properties ==
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