The set of minimal distances in Krull monoids

Abstract

Let $H$ be a Krull monoid with finite class group $G$. Then every non-unit $a \in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L (a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$. If $G$ is finite, then there is a constant $M \in \mathbb N$ such that all sets of lengths are almost arithmetical multiprogressions with bound $M$ and with difference $d \in \Delta^* (H)$, where $\Delta^* (H)$ denotes the set of minimal distances of $H$. We show that $\max \Delta^* (H) \le \max \{\exp (G)-2, \mathsf r (G)-1\}$ and that equality holds if every class of $G$ contains a prime divisor, which holds true for holomorphy rings in global fields.