On products of k atoms

Abstract

Let H be an atomic monoid. For \(k \in {\Bbb N}\) let \({\cal V}_k (H)\) denote the set of all \(m \in {\Bbb N}\) with the following property: There exist atoms (irreducible elements) u 1, …, u k , v 1, …, v m ∈ H with u 1· … · u k = v 1 · … · v m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets \({\cal V}_k (H)\) are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every \(k \in {\Bbb N}\), max \({\cal V}_{2k+1} (H) = k \vert G\vert + 1\) which settles Problem 38 in [4].