Functiones et Approximatio Commentarii Mathematici

Unions of sets of lengths

Michael Freeze and Alfred Geroldinger

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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


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