Sets of minimal distances and characterizations of class groups of Krull monoids

Qinghai Zhong

doi:10.1007/s11139-016-9873-2

Abstract

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then every non-unit 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. 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 study the structure of Delta ^* (H) and establish a characterization when Delta ^*(H) is an interval. The system mathcal L (H) = { mathsf L (a) mid a in H } of all sets of lengths depends only on the class group G, and a standing conjecture states that conversely the system mathcal L (H) is characteristic for the class group. We confirm this conjecture (among others) if the class group is isomorphic to C_n^r with r,n in mathbb N and Delta ^*(H) is not an interval.

Highlights

Introduction and main resultsLet H be a Krull monoid with finite class group G such that every class contains a prime divisor
Every non-unit of H has a factorization as a finite product of atoms, and all these factorizations are unique (i.e., H is factorial) if and only if G is trivial
There are elements having factorizations which differ up to associates and up to the order of the factors. These phenomena are described by arithmetical invariants such as sets of lengths and sets of distances

Summary

Introduction and main results

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor (holomorphy rings in global fields are such Krull monoids and more examples will be given later). Theorem 1.1 Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Zero-sum theoretical invariants (such as the Davenport constant or the cross number) and the associated inverse problems play a crucial role (surveys and detailed presentations of such results can be found in [8,10,17]).

Background on Krull monoids and their sets of minimal distances

Krull monoids

Zero-sum theory

Minimal non-half-factorial subsets of finite abelian groups

Proofs of the main theorems