ABSTRACT
In this chapter we begin to lay the groundwork necessary to explore those domains for which a unique factorization theorem is true. We wish to obtain a common generalization of Z and Q[x]. In both Z and Q[x], we ended up factoring elements into irreducibles.
Roughly speaking, this means elements that admit no ‘non-trivial’ factorizations. But what do we mean by ‘non-trivial’? In the case of the integers we disregarded factors of ±1, while in the ring of polynomials with coefficients from a field we disregarded scalar factors (that is, polynomials of degree 0). In either case, these trivial factors amount exactly to those elements of the domain that have multiplicative inverses; that is, the units.
