Units in Noncommutative Orders

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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.
Original languageEnglish
Pages (from-to)119-136
Number of pages18
JournalGroups, Rings and Group Rings
Issue number248
Publication statusPublished - 2006


  • units
  • orders


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