ABSTRACT
The theory of non-unique factorizations in integral domains has its origins in the theory of algebraic number fields. This chapter provides an overview of results which can be derived in finitely generated and finitely primary monoids. The language of monoids is the adequate framework for the formulation and investigation of most factorization problems, even if the main interest lies in the applications for integral domains. In many cases, the factorization properties of an integral domain have their counterparts in suitably constructed monoids and it is much easier to deal with them in those monoids instead of considering the integral domains themselves. Throughout the chapter, a monoid is assumed to be commutative and cancellative. The chapter provides a proposition that collects some simple facts showing that the arithmetic of finitely primary monoids is quite different from that of finitely generated monoids. It also makes the description of the class group of weakly Krull domains of finite type more explicit.
