Factorization in finitely generated domains

✍ Scribed by Wolfgang Hassler

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 311 KB
• Volume: 186
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(03)00142-7

✦ Synopsis

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 = u1 • : : : • un into irreducible elements ui of D. The integer n in this equation is called the length of the factorization. In general, elements of Noetherian domains have many (essentially) di erent factorizations.

In this article we study the non-uniqueness of factorizations in domains which are ÿnitely 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) ÿniteness conditions are fulÿlled. As a by product of our investigations it turns out that there exists a "transfer" homomorphism ÿ from our domain D to a certain block monoid B of some ÿnite semigroup C. We are able to show that the ÿniteness 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.