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In mathematics, a unique factorization domain (UFD) is a commutative ring in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki. Unique factorization domains appear in the following chain of class inclusions: == Definition == Formally, a unique factorization domain is defined to be an integral domain ''R'' in which every non-zero element ''x'' of ''R'' can be written as a product (an empty product if ''x'' is a unit) of irreducible elements ''p''i of ''R'' and a unit ''u'': :''x'' = ''u'' ''p''1 ''p''2 ... ''p''''n'' with ''n'' ≥ 0 and this representation is unique in the following sense: If ''q''1,...,''q''''m'' are irreducible elements of ''R'' and ''w'' is a unit such that :''x'' = ''w'' ''q''1 ''q''2 ... ''q''''m'' with ''m'' ≥ 0, then ''m'' = ''n'', and there exists a bijective map φ : → such that ''p''''i'' is associated to ''q''φ(''i'') for ''i'' ∈ . The uniqueness part is usually hard to verify, which is why the following equivalent definition is useful: :A unique factorization domain is an integral domain ''R'' in which every non-zero element can be written as a product of a unit and prime elements of ''R''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Unique factorization domain」の詳細全文を読む スポンサード リンク
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