The catenary degree of krull monoids I

Abstract

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree c(H) of H is the smallest integer N with the following property: for each a ∈ H and each two factorizations z, z' of a, there exist factorizations z = z0,., zk = z' of a such that, for each i ∈ [1, k], zi arises from zi-1 lacing atby rep most N atoms from zi-1 by at most N new atoms. Under a very mild condition on the Davenport constant of G, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between c(H) and the set of distances of H and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on c(H) and characterize when c(H) ≤ 4. © Société Arithmétique de Bordeaux.

Publication Title

Journal de Theorie des Nombres de Bordeaux

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