irreducible of a UFD is prime
Any irreducible element of a factorial ring is a prime element
of .
Proof. Let be an arbitrary irreducible element of . Thus is a non-unit. If , then with . We write as products of irreducibles:
Here, one of those first two products may me empty, i.e. it may be a unit. We have
(1) |
Due to the uniqueness of prime factorization, every factor is an associate
of certain of the irreducibles on the left hand side of (1). Accordingly, has to be an associate of one of the ’s or ’s. It means that either or . Thus, is a prime ideal
of , and its generator
must be a prime element.