ABSTRACT
One way to view group of divisibility is as the multiplicative group of principal fractional ideals of an integral domian partially ordered by reverse containment. Then the divisibility and factorization properties of elements of integral domian usually have their corresponding order interpretation in the group of divisibility. The study of divisibility of elements in the integral domain and the study of factorization of elements in the integral domain are related but not identical. This chapter focuses on problems that relate to the group of divisibility and on finiteness conditions on certain submonoids of an integral domain. It reviews some recent work on atomic domains for which the unique atomic factorization assumption is weakened. The chapter also determines when the monoid of nonzero fractional ideals of an integral domain is finitely generated.
