be a commutative integral domain
is a unique factorization domain
(or UFD) if
Every element of
R which is not zero or a unit can be written as a product of
irreducible elements of R.
- This expression is
unique in the sense that if we write
in R, for some nonzero non-unit c, with each ai
and bj irreducible, then we have r=s and
after renumbering the ai if necessary we have that
each ai is an associate of bj.
Of course, the fundamental theorem of arithmetic asserts that the ring
of integers Z is a UFD.
Proposition Any principal ideal domain is a UFD.
After the Proposition and the principal ideal domain writeup we know
that besides the ring of integers we have several examples
of UFDs. The ring of Gaussian integers Z[i] is a UFD and so is
the polynomial ring k[x], for any field k.
Proof of the Proposition
Let R be a PID. I first claim that every
nonzero nonunit a in R can be
factorized as product of irreducibles. If a is irreducible
there is nothing to prove. If not then we can write
a=a1b1 with a1,b1
non-units. If a1,b1 are both irreducible
we are done. Suppose that a1 is not.
Then we factorize a1=a2b2,
as a product. We repeat
this until we have expressed a as a product of irreducibles.
Thus we will have a=a1b1=a2b1b2...
We have to show that this process terminates. Write a0=a.
Then associated to our factorization attempt we have the increasing sequence of
ideals a0R < a1R < a1R < ....
Our factorisation attempt terminates if this chain terminates.
Let I be the union of these ideals. Because this sequence
is increasing it is not hard to see that I is also an ideal.
Since R is a PID this means that I=bR. Now b
has to be in one of the ideals aiR (by the definition
of I). Thus
must lie in I. It follows from this that b and
ai are associates. Clearly then ajR=aiR
for any j>=i, as required to show the existence of factorizations.
We now have to show that factorizations are unique. But we know
(since we have a PID) that the concepts of primeness and
irreducibility coincide. Thus it is easy to adapt the familiar
proof of the fundmental theorem of arithmetic to this case.