# 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 $a=u_1 \cdot \ldots \cdot u_k$, with irreducibles $u_1, \ldots u_k \in H$, 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 $H'$ with class group $G'$ such that every class contains a prime divisor and suppose that $\mathcal L (H) = \mathcal L (H')$. We show that, if one of the groups $G$ and $G'$ is finite and has rank at most two, then $G$ and $G'$ are isomorphic (apart from two well-known pairings).
Keywords :
Document type :
Preprints, Working Papers, ...
Domain :

Cited literature [38 references]

https://hal.archives-ouvertes.fr/hal-01131955
Contributor : Wolfgang Schmid <>
Submitted on : Monday, February 11, 2019 - 12:34:30 PM
Last modification on : Wednesday, March 20, 2019 - 1:15:54 AM
Long-term archiving on : Sunday, May 12, 2019 - 2:16:30 PM

### Files

systems-rank-two-groups-arxiv....
Files produced by the author(s)

### Identifiers

• HAL Id : hal-01131955, version 2
• ARXIV : 1503.04679

### Citation

Alfred Geroldinger, Wolfgang Schmid. A characterization of class groups via sets of lengths. 2018. ⟨hal-01131955v2⟩

Record views