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The set of minimal distances in Krull monoids

Volume 173 / 2016

Alfred Geroldinger, Qinghai Zhong Acta Arithmetica 173 (2016), 97-120 MSC: 11B30, 11R27, 13A05, 20M13. DOI: 10.4064/aa7906-1-2016 Published online: 11 May 2016

Abstract

Let $H$ be a Krull monoid with class group $G$. Then every nonunit $a \in H$ 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.

Authors

  • Alfred GeroldingerInstitute for Mathematics and Scientific Computing University of Graz
    NAWI Graz
    Heinrichstraße 36
    8010 Graz, Austria
    e-mail
  • Qinghai ZhongInstitute for Mathematics and Scientific Computing University of Graz
    NAWI Graz
    Heinrichstraße 36
    8010 Graz, Austria
    e-mail

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