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

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 $a\in H$ can be written as a product of irreducible elements, say $a=u_1\dots u_k$, 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 $\mathcal{L}(H)=\{\mathsf{L}(a)\mid a\in H\}$ 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 $H^{\prime }$ be a further transfer Krull monoid over a finite ablian group $G^{\prime }$ and suppose that $\mathcal{L}(H)=\mathcal{L}(H^{\prime })$. We prove that, if $G\cong C_n^r$ with $r\le n-3$ or ($r\ge n-1\ge 2$ and $n$ is a prime power), then $G$ and $G^{\prime }$ are isomorphic.

arXiv commentto appear in Communications in Algebra

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