Arithmetical invariants of local quaternion orders
Baeth, Nicholas R. ; Smertnig, Daniel
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 ▫$\rho(R^\bullet) = \sup\{k/l : ...u_1\cdots u_k = v_1 \cdots v_l$▫ with ▫$u_i$▫, ▫$v_j$▫ atoms of ▫$R^\bullet$▫ 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.
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