The set of minimal distances in Krull monoids

Alfred Geroldinger¹ ; Qinghai Zhong¹

¹ Institute for Mathematics and Scientific Computing University of Graz NAWI Graz Heinrichstraße 36 8010 Graz, Austria

Acta Arithmetica, Tome 173 (2016) no. 2, pp. 97-120

Résumé

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.

Détail

DOI : 10.4064/aa7906-1-2016

Keywords: krull monoid class group every nonunit written finite product atoms say cdot ldots cdot set mathsf possible factorization lengths called set lengths finite there constant mathbb sets lengths almost arithmetical multiprogressions bound difference delta * where delta * denotes set minimal distances max delta * max exp mathsf equality holds every class contains prime divisor which holds holomorphy rings global fields