ABSTRACT
You should now recall the proof of the uniqueness of factorization into irreducibles for Z (or Q[x]). Our intent in this chapter is to construct a more general context in which this proof is true.
This proof relies heavily on the fact that irreducible elements in Z (or Q[x]) are in fact prime. It should not then be surprising that a general unique factorization theorem should rely on the same considerations. We now define prime elements in an arbitrary domain: A non-unit p 6= 0 of a domain R is a prime if, whenever p divides ab, then p divides a or b.
