Sets of lengths in maximal orders in central simple algebras

Abstract

Let O be a holomorphy ring in a global field K, and R a classical maximal 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 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 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.