Finiteness of elasticities of orders in central simple algebras
Volume 206 / 2022
Abstract
Let $\mathcal O$ be an order in a central simple algebra $A$ over a number field. The elasticity $\rho (\mathcal O)$ is the supremum of all fractions $k/l$ such that there exists a non-zero-divisor $a \in \mathcal O$ that has factorizations into atoms (irreducible elements) of lengths $k$ and $l$. We characterize the finiteness of the elasticity for Hermite orders $\mathcal O$ if either $\mathcal O$ is a quaternion order, or $\mathcal O$ is an order in a central simple algebra of larger dimension and $\mathcal O_{\mathfrak p}$ is a tiled order at every finite place $\mathfrak p$ at which $A_{\mathfrak p}$ is not a division ring. We also prove a transfer result for such orders. This extends previous results for hereditary orders to a non-hereditary setting.
