Sets of minimal distances and characterizations of class groups of Krull monoids

Abstract

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. 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 $a$. 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 study the structure of ${\Delta }^{\ast }(H)$ and establish a characterization when ${\Delta }^{\ast }(H)$ is an interval. The system $\mathcal{L}(H)=\{\mathsf{L}(a)\mid a\in H\}$ of all sets of lengths depends only on the class group $G$, and a standing conjecture states that conversely the system $\mathcal{L}(H)$ is characteristic for the class group. We confirm this conjecture (among others) if the class group is isomorphic to $C_n^r$ with $r,n\in \mathbb{N}$ and ${\Delta }^{\ast }(H)$ is not an interval.

To appear in The Ramanujan Journal. arXiv admin note: substantial text overlap with arXiv:1506.05223