ABSTRACT
In the last chapter we acquainted ourselves with the constituent pieces of a general theory of factorization for domains: namely, the irreducibles. In this chapter and the next we discuss what is required to obtain a factorization into irreducibles, and along the way we meet some vitally important concepts for all of ring theory. Conditions necessary to force such a factorization to be unique will be examined in Chapter 13. You should now recall the proof that every integer can be factored
into irreducibles (and the analogous proof for Q[x]). Both of these proofs depend heavily on the fact that N is well ordered: by continuing to extract factors from a positive integer, we decrease its size, and we cannot continue this indefinitely. What more general context is possible?
