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单词 UFD
释义

UFD


An integral domainMathworldPlanetmath D satisfying

  • Every nonzero element of D that is not a unit can be factored into a product of a finite number of irreduciblesPlanetmathPlanetmathPlanetmath,

  • If  p1p2pr  and  q1q2qs  are two factorizations of the same element a into irreducibles, then  r=s  and we can reorder the qj’s in a way that qj is an associateMathworldPlanetmath element of pj for all j

is called a unique factorization domainMathworldPlanetmath (UFD), also a factorial ring.

The factors  p1,p2,,pr  are called the prime factorsMathworldPlanetmath of a.

Some of the classic results about UFDs:

  • On a UFD, the concept of prime elementMathworldPlanetmath and irreducible element coincide.

  • If F is a field, then F[x] is a UFD.

  • If D is a UFD, then D[x] (the ring of polynomials on the variable x over D) is also a UFD.

    Since  R[x,y]R[x][y],  these results can be extended to rings of polynomials with a finite number of variables.

  • If D is a principal ideal domainMathworldPlanetmath, then it is also a UFD.

    The converse is, however, not true.  Let F a field and consider the UFD F[x,y].Let I the ideal consisting of all the elements of F[x,y] whose constant term is 0.  Then it can be proved that I is not a principal idealMathworldPlanetmathPlanetmathPlanetmath.  Therefore not every UFD is a PID.

TitleUFD
Canonical nameUFD
Date of creation2013-03-22 11:56:22
Last modified on2013-03-22 11:56:22
Ownerdrini (3)
Last modified bydrini (3)
Numerical id19
Authordrini (3)
Entry typeDefinition
Classificationmsc 13G05
Synonymunique factorization domain
Related topicIntegralDomain
Related topicIrreducible
Related topicEuclideanRing
Related topicEuclideanValuation
Related topicProofThatAnEuclideanDomainIsAPID
Related topicWhyEuclideanDomains
Related topicY2X32
Related topicPID
Related topicPIDsAreUFDs
Related topicFundamentalTheoremOfArithmetic
Definesfactorial ring
Definesprime factor
DefinesUFD
Definesunique factorization
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更新时间:2025/5/4 14:14:11