Editing Unique factorization domain (section)

{{Short description|Type of integral domain}}
{{Redirect|Unique factorization|the uniqueness of integer factorization|fundamental theorem of arithmetic}}
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{{Algebraic structures |Ring}}
In [[mathematics]], a '''unique factorization domain''' ('''UFD''') (also sometimes called a '''factorial ring''' following the terminology of [[Nicolas Bourbaki|Bourbaki]]) is a [[Ring (mathematics)|ring]] in which a statement analogous to the [[fundamental theorem of arithmetic]] holds. Specifically, a UFD is an [[integral domain]] (a [[zero ring|nontrivial]] [[commutative ring]] in which the product of any two non-zero elements is non-zero) in which every non-zero non-[[Unit (ring theory)|unit]] element can be written as a product of [[irreducible element]]s, uniquely up to order and units.

Important examples of UFDs are the integers and [[polynomial ring]]s in one or more variables with coefficients coming from the integers or from a [[Field (mathematics)|field]].

Unique factorization domains appear in the following chain of [[subclass (set theory)|class inclusions]]:
{{Commutative ring classes}}