Arithmetical invariants of local quaternion orders

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 : u_1\cdots u_k = v_1 \cdots v_l \text{ with $u_i$,$v_j$atomsof$R^\bullet$and$k$,$l \ge 1$} \,\}$ and is one of the basic arithmetical invariants that is studied in factorization theory. We characterize finiteness of $\rho (R^{\bullet })$ and show that the set of distances $\Delta (R^{\bullet })$ 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.

33 pages; final revision (some typos fixed)

References in corpus (1)

• Factorizations of Elements in Noncommutative Rings: A Survey

Cited by in corpus (1)

• On strongly primary monoids, with a focus on Puiseux monoids