Unique factorization ___domain: Difference between revisions

Revision as of 07:01, 29 July 2008

Revision as of 00:16, 2 August 2008

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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.g.:
:: <math>\sin \pi z = \pi z \prod_{n=1}^{\infty} \left(1-{{z^2}\over{n^2}}\right).</math>

== Properties ==