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.

Title UFD
Canonical name UFD
Date of creation 2013-03-22 11:56:22
Last modified on 2013-03-22 11:56:22
Owner drini (3)
Last modified by drini (3)
Numerical id 19
Author drini (3)
Entry type Definition
Classification msc 13G05
Synonym unique factorization domain
Related topic IntegralDomain
Related topic Irreducible
Related topic EuclideanRing
Related topic EuclideanValuation
Related topic ProofThatAnEuclideanDomainIsAPID
Related topic WhyEuclideanDomains
Related topic Y2X32
Related topic PID
Related topic PIDsAreUFDs
Related topic FundamentalTheoremOfArithmetic
Defines factorial ring
Defines prime factor
Defines UFD
Defines unique factorization