A characterization of class groups via sets of lengths
J. Korean Math. Soc. 2019 Vol. 56, No. 4, 869-915
Published online July 1, 2019
Alfred Geroldinger, Wolfgang Alexander Schmid
University of Graz, Universit{\'e} Paris 13
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 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 $G$. We study the inverse question asking whether the system $\mathcal L (H)$ is characteristic for the class group. Let $H'$ be a further Krull monoid 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 exceptions).
Keywords : Krull monoids, class groups, arithmetical characterizations, sets of lengths, zero-sum sequences
MSC numbers : 11B30, 11R27, 13A05, 13F05, 20M13
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