Krull Domains and Monoids, Their Sets of Lengths, and Associated Combinatorial Problems

This chapter provides a discussion on the theory of non-unique factorization. It considers factorization properties of Krull domains, including integrally closed noetherian domains and rings of integers in algebraic number fields. The chapter develops the algebraic properties of Krull monoids and provides a wide array of examples. It discusses sets of lengths and defines several combinatorial constants which play a key role in describing the arithmetic of a Krull monoid. Quantitative investigations of phenomena of non-unique factorizations are interesting mainly for holomorphy rings in global fields (including rings of integers in algebraic number fields and in algebraic function fields in one variable over a finite field). However, it has turned out that most of the results can be derived for very general structures in the setting of abstract analytic number theory. The chapter also discusses combinatorial problems in abelian groups.