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 and 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 are called the prime factors of .
Some of the classic results about UFDs:
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 , 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 . Let the ideal consisting of all the elements of whose constant term is . Then it can be proved that is not a principal ideal. Therefore not every UFD is a PID.
|Date of creation||2013-03-22 11:56:22|
|Last modified on||2013-03-22 11:56:22|
|Last modified by||drini (3)|
|Synonym||unique factorization domain|