#### Title

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

#### Recommended Citation

Geroldinger, A., Grynkiewicz, D., & Schmid, W.
(2011). The catenary degree of krull monoids I.* Journal de Theorie des Nombres de Bordeaux**, 23* (1), 137-169.
https://doi.org/10.5802/jtnb.754