You are here
HomeUFD
Primary tabs
UFD
An integral domain $D$ satisfying

Every nonzero element of $D$ that is not a unit can be factored into a product of a finite number of irreducibles,
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 $a$.
Some of the classic results about UFDs:

On a UFD, the concept of prime element 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 $R[x,\,y]\cong R[x][y]$, these results can be extended to rings of polynomials with a finite number of variables.

If $D$ is a principal ideal domain, 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 ideal. Therefore not every UFD is a PID.
Mathematics Subject Classification
13G05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections
Attached Articles
Corrections
UDF by perucho ✓
spelling by Mathprof ✓
Missing period by Mathprof ✓
missing word(s) by Wkbj79 ✓