Let $A$ be a finite-dimensional algebra over the rational number field $\Bbb Q$. A subring $\Gamma$ with the same unit element is called an order if $\Gamma$ is a finitely generated $\Bbb Z$-submodule such that $\Gamma$ contains a $\Bbb Q$-basis of $A$. Although the unit group $U(\Gamma)$ of $\Gamma$ is finitely generated, the determination of a finite set of generators seems to be a problem beyond reach. The authors give a survey of recent accomplishments on the following topics concerning $U(\Gamma)$: (1) special subgroups; (2) generators for a subgroup of finite index; (3) orders in quaternion algebras.
Dooms, A & Jespers, E 2006, 'Units in Noncommutative Orders', Groups, Rings and Group Rings, no. 248, pp. 119-136.
Dooms, A., & Jespers, E. (2006). Units in Noncommutative Orders. Groups, Rings and Group Rings, (248), 119-136.
@article{5bde6ec019da40a99f84f60118fdf846,
title = "Units in Noncommutative Orders",
abstract = "Let $A$ be a finite-dimensional algebra over the rational number field $\Bbb Q$. A subring $\Gamma$ with the same unit element is called an order if $\Gamma$ is a finitely generated $\Bbb Z$-submodule such that $\Gamma$ contains a $\Bbb Q$-basis of $A$. Although the unit group $U(\Gamma)$ of $\Gamma$ is finitely generated, the determination of a finite set of generators seems to be a problem beyond reach. The authors give a survey of recent accomplishments on the following topics concerning $U(\Gamma)$: (1) special subgroups; (2) generators for a subgroup of finite index; (3) orders in quaternion algebras.",
keywords = "units, orders",
author = "Ann Dooms and Eric Jespers",
year = "2006",
language = "English",
pages = "119--136",
journal = "Groups, Rings and Group Rings",
number = "248",
}