Non-Atomic Unique Factorization in Integral Domains
Unique factorization domains are of course the integral domains in which every nonzero nonunit element has a unique factorization (up to order and associates) into irreducible elements or atoms. Now UFDs can also be characterized by the property that every nonzero nonunit is a product of principal primes or equivalently that every nonzero nonunit has the form upa11 · · · pann where u is a unit, p1, . . . , pn are nonassociate principal primes, and each ai ≥ 1. Each of the paii , in addition to being a power of a prime, has other properties, each of which is subject to generalization. For example, each paii is primary, each is contained in a unique maximal t-ideal, and the paii are pairwise coprime. The goal of this chapter is to survey various generalizations of (unique) factorization into prime powers in integral domains. This follows the thesis of M. Zafrullah that the paii are the building blocks in a UFD. The author would like to thank M. Zafrullah for a number of discussions of these topics over the past several years.