Arithmetical invariants of local quaternion orders

2017/06/02 by Baeth, Nicholas R., Smertnig, Daniel
#11R27 #11S45 #16H10 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1706.00572

Abstract

Let D be a DVR, let K be its quotient field, and let R be a D-order in a quaternion algebra A over K. The elasticity of R^\bullet is ρ(R^\bullet) = sup\ k/l : u₁⋯ u_k = v₁ ⋯ v_l with u_i, v_j atoms of R^\bullet and k, l ≥ 1 \ and is one of the basic arithmetical invariants that is studied in factorization theory. We characterize finiteness of ρ(R^\bullet) and show that the set of distances Δ(R^\bullet) and all catenary degrees \mathsf c_\mathsf d(R^\bullet) are finite. In the setting of noncommutative orders in central simple algebras, such results have only been understood for hereditary orders and for a few individual examples.

Related

• Elasticities of Orders in Central Simple Algebras
• Elasticity in orders of an algebraic number field with radical conductor ideal and their rings of formal power series
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• Constructing non-maximal orders in quaternion algebras
• The Dedekind-Hasse Criterion in Quaternion Algebras