Sets of lengths in maximal orders in central simple algebras

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.

40 pages; final version, with minor edits over previous one

References in corpus (3)

• The Catenary Degree of Krull Monoids I
• A construction of integer-valued polynomials with prescribed sets of lengths of factorizations
• Non-commutative Krull monoids: A divisor theoretic approach and their arithmetic

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• Factorizations in upper triangular matrices over information semialgebras
• Factorizations of Elements in Noncommutative Rings: A Survey
• Sets of Arithmetical Invariants in Transfer Krull Monoids
• The catenary degree of Krull monoids II
• Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules
• A Characterization of class groups via sets of lengths {II}
• Long sets of lengths with maximal elasticity
• A characterization of class groups via sets of lengths
• Sets of minimal distances and characterizations of class groups of Krull monoids
• On noncommutative bounded factorization domains and prime rings
• On congruence half-factorial Krull monoids with cyclic class group
• On the Algebraic and Arithmetic structure of the monoid of Product-one sequences II
• On half-factoriality of transfer Krull monoids
• Definite orders with locally free cancellation
• On transfer homomorphisms in commutative rings with zero-divisors
• Factorization in monoids and rings
• Factorization Theory in Commutative Monoids
• Factorization in Finitely-Presented Monoids
• On transfer Krull monoids
• Elasticities of Orders in Central Simple Algebras
• A characterization of finite abelian groups via sets of lengths in transfer Krull monoids