PlanetMath: UFD

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Owner confidence rating: High

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UFD

(Definition)

An integral domain satisfying

Every nonzero element of that is not a unit can be factored into a product of a finite number of irreducibles,
If $p_1p_2\cdots p_r$ and $q_1q_2\cdots q_s$ are two factorizations of the same element into irreducibles, then and we can reorder the 's in a way that is an associate element of for all

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

The factors $p_1,\,p_2,\,\ldots,\,p_r$ are called the prime factors of .

Some of the classic results about UFDs:

On a UFD, the concept of prime element and irreducible element coincide.
If is a field, then is a UFD.
If is a UFD, then (the ring of polynomials on the variable over ) is also a UFD.
Since $R[x,\,y]\cong R[x][y]$ , these results can be extended to rings of polynomials with a finite number of variables.
If is a principal ideal domain, then it is also a UFD.
The converse is however, not true. Let a field and consider the UFD $F[x,\,y]$ . Let the ideal consisting of all the elements of $F[x,\,y]$ whose constant term is 0. Then it can be proved that is not a principal ideal. Therefore not every UFD is a PID.

"UFD" is owned by drini. [ full author list (3) | owner history (1) ]

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See Also: integral domain, irreducible, Euclidean domain, Euclidean valuation, proof that a Euclidean domain is a PID, motivation for Euclidean domains,

$y^2= x^3-2$

, PID, every PID is a UFD, fundamental theorem of arithmetic

Other names:

unique factorization domain

Also defines:

factorial ring, prime factor

Keywords:

Ring, Domain, Factorization

Attachments:: example of ring which is not a UFD (Example) by alozano
prime factors of (Result) by pahio
irreducible of a UFD is prime (Theorem) by pahio

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Cross-references: principal ideal, constant term, ideal, converse, principal ideal domain, variable, polynomials, ring, field, irreducible element, prime element, factors, associate, irreducibles, number, finite, product, unit, integral domain
There are 43 references to this entry.

This is version 13 of UFD, born on 2001-11-04, modified 2008-05-22.
Object id is 671, canonical name is UFD.
Accessed 9846 times total.

Classification:

13G05 (Commutative rings and algebras :: Integral domains)

Pending Errata and Addenda

None.[ View all 6 ]

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