Unique Factorization Domain
In mathematics, a unique factorization domain (UFD) is a commutative ring in which every non-unit element, with special exceptions, can be uniquely written as a product of prime elements (or irreducible elements), analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki.
Note that unique factorization domains appear in the following chain of class inclusions:
- Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields
Other articles related to "unique, factorization, unique factorization domains, domain, unique factorization domain, domains":
... is less than the degree of q, and a decomposition with these properties is unique ... Rings for which there exists unique (in an appropriate sense) factorization of nonzero elements into irreducible factors are called unique factorization domains or factorial rings the given construction shows that all ... Thus the polynomial ring K is a principal ideal domain, and for the same reason every Euclidean domain is a principal ideal domain ...
... A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal ... Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial ... In this case it is in fact a principal ideal domain ...
... A unique factorization domain is a GCD domain ... Among the GCD domains, the unique factorization domains are precisely those that are also atomic domains (which means that at least one factorization into irreducible elements exists for any nonzero nonunit) ... A Bézout domain (i.e ...
... If R is an integral domain, an element f of R which is neither zero nor a unit is called irreducible if there are no non-units g and h with f = gh ... is irreducible the converse is not true in general but holds in unique factorization domains ... The polynomial ring F over a field F (or any unique-factorization domain) is again a unique factorization domain ...
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