Editing Unique factorization domain (section)

=== Non-examples ===
* The [[quadratic integer ring]] <math>\mathbb Z[\sqrt{-5}]</math> of all [[complex number]]s of the form <math>a+b\sqrt{-5}</math>, where ''a'' and ''b'' are integers, is not a UFD because 6 factors as both 2×3 and as <math>\left(1+\sqrt{-5}\right)\left(1-\sqrt{-5}\right)</math>. These truly are different factorizations, because the only units in this ring are 1 and &minus;1; thus, none of 2, 3, <math>1+\sqrt{-5}</math>, and <math>1-\sqrt{-5}</math> are [[Unit (ring theory)|associate]]. It is not hard to show that all four factors are irreducible as well, though this may not be obvious.{{sfnp|Artin|2011|p=360|ps=}} See also ''[[Algebraic integer]]''.
* For a [[Square-free integer|square-free positive integer]] ''d'', the [[ring of integers]] of <math> \mathbb Q[\sqrt{-d}]</math> will fail to be a UFD unless ''d'' is a [[Heegner number]].
* The ring of formal power series over the complex numbers is a UFD, but the [[subring]] of those that converge everywhere, in other words the ring of [[entire function]]s in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be finite, e.g.:
*: <math>\sin \pi z = \pi z \prod_{n=1}^{\infty} \left(1-{{z^2}\over{n^2}}\right).</math>