About: Unique factorization domain

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In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) 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. Unique factorization domains appear in the following chain of class inclusions:

Attributes	Values
rdf:type	Thing anatomical structure
rdfs:label	Unique factorization domain (en)
rdfs:comment	In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) 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. Unique factorization domains appear in the following chain of class inclusions: (en)
sameAs	Unique factorization domain Unique factorization domain
dbp:wikiPageUsesTemplate	dbt:Algebraic_structures dbt:Authority_control dbt:Citation dbt:Cite_book dbt:Cite_journal dbt:Harv dbt:Redirect dbt:Refbegin dbt:Refend dbt:Reflist dbt:Lang_Algebra dbt:Commutative_ring_classes
Subject	Algebraic number theory Ring theory
gold:hypernym	Ring
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Unique_factorization_domain?oldid=1074152312&ns=0
Wikipage page ID	60546 (xsd:integer)
page length (characters) of wiki page	13313 (xsd:nonNegativeInteger)
Wikipage revision ID	1074152312 (xsd:integer)
has abstract	In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) 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. Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field. Unique factorization domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Unique_factorization_domain
is Wikipage redirect of	Unique Factorization Domain Unique factorisation domain Factorial domain Factorial ring UFD (math) Unique factorisation Unique factorization
is Link from a Wikipage to another Wikipage of	Fractional ideal Greatest common divisor List of unsolved problems in mathematics Noetherian ring Noncommutative unique factorization domain 11 (number) 163 (number) Algebraic geometry of projective spaces Algebraic number field Algebraic number theory Arithmetic derivative Ascending chain condition on principal ideals Atomic domain Auslander–Buchsbaum theorem Berlekamp's algorithm Binary GCD algorithm Bézout domain Cantor–Zassenhaus algorithm Germ (mathematics) Glossary of algebraic geometry Glossary of commutative algebra Glossary of ring theory Discrete valuation ring Divisor (algebraic geometry) Domain Hilbert class field Cubic reciprocity Cyclotomic field Dedekind domain List of number fields with class number one Commutative ring Complex multiplication Eisenstein's criterion Hensel's lemma Hilbert's Nullstellensatz Cubic function List of commutative algebra topics Finite field Fundamental theorem of arithmetic GCD domain Gauss's lemma (polynomials) Gaussian integer Ideal class group Factorial domain Factorial ring Exponential polynomial Factorization Factorization of polynomials Fermat's Last Theorem Fermat's factorization method Thue's lemma Heegner number Integral domain Integrally closed domain Irrational number Irreducible element Irreducible fraction Irreducible polynomial Krull dimension Krull ring Lagrange's four-square theorem Euclid's lemma Euclidean algorithm Euclidean domain Iwasawa algebra Kleinian integer

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