Let \$A\$ be a finite-dimensional algebra over the rational number field \$\textbackslash{}Bbb Q\$. A subring \$\textbackslash{}Gamma\$ with the same unit element is called an order if \$\textbackslash{}Gamma\$ is a finitely generated \$\textbackslash{}Bbb Z\$-submodule such that \$\textbackslash{}Gamma\$ contains a \$\textbackslash{}Bbb Q\$-basis of \$A\$. Although the unit group \$U(\textbackslash{}Gamma)\$ of \$\textbackslash{}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(\textbackslash{}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 \$\textbackslash{}Bbb Q\$. A subring \$\textbackslash{}Gamma\$ with the same unit element is called an order if \$\textbackslash{}Gamma\$ is a finitely generated \$\textbackslash{}Bbb Z\$-submodule such that \$\textbackslash{}Gamma\$ contains a \$\textbackslash{}Bbb Q\$-basis of \$A\$. Although the unit group \$U(\textbackslash{}Gamma)\$ of \$\textbackslash{}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(\textbackslash{}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",
}