The investigation of properties of integral domains, in particular rings of integers of algebraic number fields that may not be unique factorization domains, is the original motivation for the study of the phenomena of non-unique factorization. It turns out that the phenomena are of a purely multiplicative nature and therefore we can restrict ourselves to the study of multiplicative monoids of integral domains. This strategy is pursued in this monograph, which is mainly concerned with the non-unique factorization properties of commutative cancellative monoids.

Chapter 1 surveys basic classical notions of the theory of non-unique factorization together with some elementary factorization properties of the rings of integers of algebraic number fields. Various invariants serving to classify the non-unicity of factorizations (sets of lengths, elasticity, catenary degree and tame degree) are introduced. Chapter 2 is an introduction to the theory of non-unique factorization of commutative cancellative monoids. The theory of v-ideals is developed alongside definitions of some auxiliary monoids. At the end of the chapter, results obtained for these monoids are applied to a study of factorization properties of integral domains. Chapter 3 is devoted to a study of arithmetic properties of the auxiliary monoids introduced in the previous chapter and again the results obtained are applied to integral domains. Chapter 4 deals with sets of lengths of factorizations. Under a rather general assumption on a monoid, the structure of its set of length is described.

Chapter 5 is a self-contained introduction to additive group theory. Its results are applied in chapters 6 and 7. Krull monoids with finite class group with the additional property that every class contains a prime are studied. These monoids are of particular interest because they include multiplicative monoids of integers of algebraic number fields and of holomorphy rings in algebraic function fields over finite fields. Chapter 8 is a self-contained introduction to analytic number theory focusing on notions applied in a modern treatment of the analytic theory of non-unique factorization presented in the last chapter. There, some asymptotic formulas for various counting functions are derived and these results are applied in orders in algebraic number fields and in holomorphy rings in algebraic function fields over finite fields. The monograph deals with the phenomena of non-unique factorization naturally appearing in the most fundamental questions of algebra. Combining methods of various branches of mathematics, it brings together a theory from classical results to topics reflecting the recent ideas. It is a nice book written in a precise, readable style.

Reviewer:

pruz