Unique factorization ___domain: Difference between revisions

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In [[mathematics]], a '''unique factorization ___domain (UFD)''' is, roughly speaking, a [[commutative ring]] in which every element, with special exceptions, can be uniquely written as a product of [[prime element]]s, analogous to the [[fundamental theorem of arithmetic]] for the [[integer]]s. UFDs are sometimes called '''factorial rings''', following the terminology of [[Nicolas Bourbaki|Bourbaki]].

SomeA unique factorization ___domain is a specific kindstype of uniqueintegral factorization___domain, domainsand arecan givenbe withcharacterized by the following (not necessarily exhaustive) chain of [[subset|subclass (set theory)|class inclusions]]:

'''unique factorization domains''' ⊃ '''[[principal ideal ___domain]]s''' ⊃ '''[[Euclidean ___domain]]s''' ⊃ '''[[field (mathematics)|field]]s'''

* '''[[integral ___domain]]s''' ⊃ '''[[unique factorization domains___domain]]s''' ⊃ '''[[principal ideal ___domain]]s''' ⊃ '''[[Euclidean ___domain]]s''' ⊃ '''[[field (mathematics)|field]]s'''

== Definition ==