Sets of lengths in maximal orders in central simple algebras

Abstract

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

Factorization theory: From commutative to noncommutative settings

“…In the noncommutative setting, (weak) transfer homomorphisms to appropriate commutative semigroups have already been used to study sets of lengths in [BPA + 11,BBG14,Ger13,Sme13].…”

“…Developing our machinery in this abstract setting allows us to simultaneously treat normalizing and commutative Krull monoids as well as bounded Krull rings in the sense of Chamarie (see [Cha81,MVO12]), and in particular the classical maximal orders in central simple algebras over global fields (see [Rei75]) to which we ultimately apply our abstract results. Therefore we recall the following, referring to [Sme13] for more details. Let Q be a quotient semigroup (that is, a semigroup in which every cancellative element is invertible).…”

“…Nonetheless, rigid factorizations have been studied in the following settings: Generalizing the study of PIDs, the study of 2-firs (see [Coh85,Chapter 3]) and, on an ideal-theoretic level, saturated subcategories of arithmetical groupoids (see [Sme13]). The study of polynomial decompositions, that is, the study of the factorization properties of the noncommutative semigroup (K[X] \ K, •) where K is (usually) a field, also concerns itself with what amounts to rigid factorizations (see [ZM08]).…”

“…In [BPA + 11], [BBG14], [Ger13], and [Sme13], the focus on noncommutative factorizations was solely on the sets of lengths of factorizations of a given element; that is, sets of the form L(a) = { n ∈ N 0 : a = u 1 · · · u n with each u i an atom in S } and associated invariants. Much of this work was done through the use of various generalizations of transfer homomorphisms to the noncommutative settings, each of which preserves sets of lengths.…”

“…Finally, in Section 7, we investigate catenary degrees in saturated subcategories of arithmetical groupoids and arithmetical maximal orders in quotient semigroups (as studied in [Sme13]). This treatment also includes normalizing Krull monoids considered in [Ger13].…”