A characterization of class groups via sets of lengths

Abstract

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a\in H$ can be written as a finite product of irreducible elements. If $a=u\_1\cdot \dots \cdot u\_k$, with irreducibles $u\_1,\dots u\_k\in H$, then $k$ is called the length of the factorization and the set $\mathsf{L}(a)$ of all possible $k$ is called the set of lengths of $a$. It is well-known that the system $\mathcal{L}(H)=\{\mathsf{L}(a)\mid a\in H\}$ depends only on the class group $G$. In the present paper we study the inverse question asking whether or not the system $\mathcal{L}(H)$ is characteristic for the class group. Consider a further Krull monoid $H^{\prime }$ with class group $G^{\prime }$ such that every class contains a prime divisor and suppose that $\mathcal{L}(H)=\mathcal{L}(H^{\prime })$. We show that, if one of the groups $G$ and $G^{\prime }$ is finite and has rank at most two, then $G$ and $G^{\prime }$ are isomorphic (apart from two well-known pairings).

The current version is close to the one to appear in J. Korean Math. Soc., yet it contains a detailed proof of Proposition 2.4. The content of Chapter 4 of the first version had been split off and is presented in ' A characterization of Krull monoids for which sets of lengths are (almost) arithmetical progressions' by the same authors (see hal-01976941 and arXiv:1901.03506)

Cited by in corpus (4)

• A Characterization of class groups via sets of lengths {II}
• Long sets of lengths with maximal elasticity
• Factorization invariants in numerical monoids
• Factorization Theory in Commutative Monoids