A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

2017/11/15 by Qinghai Zhong, Zhong, Qinghai · 1 citation
Mathematics · #11B30 #11R27 #13A05 #13F05 #20M13 #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1711.05437

openalex publication_date 2017/11/15 · openalex created_date 2017/12/04 · openalex updated_date 2026/07/28

Abstract

Let H be a transfer Krull monoid over a finite ablian group G (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit a ∈ H can be written as a product of irreducible elements, say a = u₁ … u_k, and the number of factors k is called the length of the factorization. The set \mathsf L (a) of all possible factorization lengths is the set of lengths of a. It is classical that the system \mathcal L (H) = \ \mathsf L (a) | a ∈ H \ of all sets of lengths depends only on the group G, and a standing conjecture states that conversely the system \mathcal L (H) is characteristic for the group G. Let H' be a further transfer Krull monoid over a finite ablian group G' and suppose that \mathcal L (H)= \mathcal L (H'). We prove that, if G≅ C_n^r with r≤ n-3 or (r≥ n-1≥ 2 and n is a prime power), then G and G' are isomorphic.

Cited by

• The set of minimal distances of the monoid of plus-minus weighted zero-sum sequences and applications to the characterization problem

Related

• Sets of Lengths
• Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids
• A characterization of class groups via sets of lengths
• The system of sets of lengths in Krull monoids under set addition
• On a zero-sum problem arising from factorization theory