# Revista Matemática Iberoamericana

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**Volume 32, Issue 2, 2016, pp. 571–588**

**DOI: 10.4171/RMI/895**

Published online: 2016-06-08

The system of sets of lengths in Krull monoids under set addition

Alfred Geroldinger^{[1]}and Wolfgang A. Schmid

^{[2]}(1) Universität Graz, Austria

(2) Université Paris 13, Villetaneuse, France

Let $H$ be a Krull monoid with class group $G$ and suppose that each class contains a prime divisor. Then every element $a \in H$ has a factorization into irreducible elements, and the set $\mathsf L (a)$ of all possible factorization lengths for $a$ is the set of lengths of $a$. We consider the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths, and we characterize (in terms of the class group $G$) when $\mathcal L (H)$ is additively closed under set addition.

*Keywords: *Krull monoids, sets of lengths, zero-sum sequences, minimal sets of distances, maximal orders

Geroldinger Alfred, Schmid Wolfgang: The system of sets of lengths in Krull monoids under set addition. *Rev. Mat. Iberoamericana* 32 (2016), 571-588. doi: 10.4171/RMI/895