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 $a=u_1\cdot \dots \cdot u_k$. 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 $d\in {\Delta }^{\ast }(H)$, where ${\Delta }^{\ast }(H)$ denotes the set of minimal distances of $H$. We show that $\max {\Delta }^{\ast }(H)\le \max \{\exp (G)-2,\mathsf{r}(G)-1\}$ and that equality holds if every class of $G$ contains a prime divisor, which holds true for holomorphy rings in global fields.

Cited by in corpus (2)

• A Characterization of class groups via sets of lengths {II}
• On half-factoriality of transfer Krull monoids