Factorization in finitely generated domains

Abstract

Let D be a Noetherian domain. Then it is well known that D is atomic, i.e. every non-zero non-unit a∈ D possesses a factorization a=u 1·…·u n into irreducible elements u i of D. The integer n in this equation is called the length of the factorization. In general, elements of Noetherian domains have many (essentially) different factorizations. In this article we study the non-uniqueness of factorizations in domains which are finitely generated Z -algebras. We investigate several arithmetical invariants (such as the catenary degree, tame degrees and sets of lengths) which are well studied in the one-dimensional case. We prove that these invariants behave similar in the higher-dimensional case if certain (natural) finiteness conditions are fulfilled. As a by product of our investigations it turns out that there exists a “transfer” homomorphism β from our domain D to a certain block monoid B of some finite semigroup C . We are able to show that the finiteness of all arithmetical invariants we study carries over from B to D. Moreover, the system of sets of lengths of D coincides with that of B .