## Functiones et Approximatio Commentarii Mathematici

- Funct. Approx. Comment. Math.
- Volume 39, Number 1 (2008), 149-162.

### Unions of sets of lengths

Michael Freeze and Alfred Geroldinger

#### Abstract

Let $H$ be a Krull monoid such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). For $k \in \mathbb{N}$ let $\mathcal{V}_k (H)$ denote the set of all $m \in \mathbb{N}$ \ with the following property{\rm \,:} There exist atoms (irreducible elements) \ $u_1, \ldots, u_k, v_1, \ldots, v_m \in H$ with $u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_m$. We show that the sets $\mathcal{V}_k (H)$ are intervals for all $k \in \mathbb{N}$. This solves Problem 37 in [4].

#### Article information

**Source**

Funct. Approx. Comment. Math., Volume 39, Number 1 (2008), 149-162.

**Dates**

First available in Project Euclid: 19 December 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.facm/1229696561

**Digital Object Identifier**

doi:10.7169/facm/1229696561

**Mathematical Reviews number (MathSciNet)**

MR2490095

**Zentralblatt MATH identifier**

1228.20046

**Subjects**

Primary: 11R27: Units and factorization

Secondary: 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 13A05: Divisibility; factorizations [See also 13F15] 20M14: Commutative semigroups

**Keywords**

non-unique factorizations sets of lengths Krull monoids

#### Citation

Freeze, Michael; Geroldinger, Alfred. Unions of sets of lengths. Funct. Approx. Comment. Math. 39 (2008), no. 1, 149--162. doi:10.7169/facm/1229696561. https://projecteuclid.org/euclid.facm/1229696561