## Integral Domains, Unique Factorization Domain

### Unique Factorization Domain

A ring is a unique factorization domain (ufd)
if it is a factorization domain and all factorizations are unique.
Imagine a factorization domain where all irreducible elements are prime.
(We already know the prime elements are irreducible.)
Apply
Euclid's proof,
and the ring becomes a ufd.
Conversely, if R is a ufd,
let an irreducible element p divide ab.
Since the factorization of ab is unique, p appears somewhere in the factors of a or b,
hence p divides a or b, and p is prime.
A factorization domain is a ufd iff prime and irreducible elements coincide.