Sets of lengths in maximal orders in central simple algebras

TL;DR

This paper investigates the structure of factorizations in maximal orders of central simple algebras over global fields, establishing when transfer homomorphisms exist and analyzing the finiteness of sets of lengths.

Contribution

It demonstrates the existence of transfer homomorphisms to zero-sum sequence monoids in most cases and characterizes the cases where such homomorphisms do not exist, especially over number fields.

Findings

Transfer homomorphisms exist in most cases, linking to zero-sum sequences.

Sets of lengths are finite and well-structured when transfer homomorphisms exist.

In certain cases over number fields, no transfer homomorphism exists and invariants are infinite.

Abstract

Let $\mathcal O$ be a holomorphy ring in a global field $K$, and $R$ a classical maximal $\mathcal O$-order in a central simple algebra over $K$. We study sets of lengths of factorizations of cancellative elements of $R$ into atoms (irreducibles). In a large majority of cases there exists a transfer homomorphism to a monoid of zero-sum sequences over a ray class group of $\mathcal O$, which implies that all the structural finiteness results for sets of lengths---valid for commutative Krull monoids with finite class group---hold also true for $R$. If $\mathcal O$ is the ring of algebraic integers of a number field $K$, we prove that in the remaining cases no such transfer homomorphism can exist and that several invariants dealing with sets of lengths are infinite.