Alfred Geroldinger: On Length Sets in Krull Monoids

Alfred Geroldinger (University of Graz): On Length Sets in Krull Monoids

Let $H$ be a Krull monoid (e.g., the multiplicative monoid of nonzero elements of a Krull domain). Then every non-invertible element can be factored into irreducible elements. If $a = u_1 \cdot \ldots \cdot u_k$, with $k \in \mathbb N_0$ and irreducibles $u_1, \ldots, u_k$, then $k$ is a factorization length and the set $\mathsf L (a)$ of all factorization lengths of $a$ is called the length set of $a$. The system $\mathcal L (H) = {\mathsf L (a) \colon a \in H }$ of length sets is an infinite family of finite subsets of $\mathbb N_0$. The structure of $\mathcal L (H)$ depends only on the class group of $H$ and on the distribution of prime divisors in the classes. We provide an overview of the current state of knowledge regarding $\mathcal L(H)$ and present some new results for the case of torsion class groups.

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