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Unique factorization domain
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of <a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki</a>) is a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> in which a statement analogous to the <a href="/facts/Fundamental_theorem_of_arithmetic/EzHUwlXX">fundamental theorem of arithmetic</a> holds. Specifically, a UFD is an <a href="/facts/Integral_domain/cylt6zxW">integral domain</a> (a <a href="/facts/Zero_ring/trFjtDJ1">nontrivial</a> <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> in which the product of any two non-zero elements is non-zero) in which every non-zero non-<a href="/facts/Unit_(ring_theory)/GJSBu8Y7">unit</a> element can be written as a product of <a href="/facts/Irreducible_element/6oMIppy4">irreducible elements</a>, uniquely up to order and units. </p><p>Important examples of UFDs are the integers and <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial rings</a> in one or more variables with coefficients coming from the integers or from a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. </p><p>Unique factorization domains appear in the following chain of <a href="/facts/Subclass_(set_theory)/FjTF1nAr">class inclusions</a>: </p> <a href="/facts/Rng_(algebra)/SquNCXKe">rngs</a> ⊃ <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a> ⊃ <a href="/facts/Commutative_ring/QS1r2ovR">commutative rings</a> ⊃ <a href="/facts/Integral_domain/cylt6zxW">integral domains</a> ⊃ <a href="/facts/Integrally_closed_domain/L3kgG2q6">integrally closed domains</a> ⊃ <a href="/facts/GCD_domain/PWrf9Tkm">GCD domains</a> ⊃ unique factorization domains ⊃ <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domains</a> ⊃ <a href="/facts/Euclidean_domain/ydwCD0fz">euclidean domains</a> ⊃ <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a> ⊃ <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed fields</a>