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Subgroups of $GL_{n}(R)$ for local rings $R$

Let $ R $ be a local ring, with maximal ideal $ {\bf m} $, and residue class division ring $ R/{\bf m} = D $. Put $ A = M_n(R), n \geq 1 $, and denote by $ A^* = GL_n(R) $ the group of units of $ A $. Here we investigate some algebraic structure of subnormal and maximal subgroups of $ A^* $. For ins...

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Detalles Bibliográficos
Autores principales: Kuku, A O, Mahdavi-Hezavehi, M
Lenguaje:eng
Publicado: 2002
Materias:
XX
Acceso en línea:http://cds.cern.ch/record/645825
Descripción
Sumario:Let $ R $ be a local ring, with maximal ideal $ {\bf m} $, and residue class division ring $ R/{\bf m} = D $. Put $ A = M_n(R), n \geq 1 $, and denote by $ A^* = GL_n(R) $ the group of units of $ A $. Here we investigate some algebraic structure of subnormal and maximal subgroups of $ A^* $. For instance, when $ D $ is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of $ A^* $ are central. It is also proved that maximal subgroups of $ A^* $ are not finitely generated. Furthermore, assume that $ P $ is a nonabelian maximal subgroup of $ GL_1(R) $ such that $ P $ contains a noncentral soluble normal subgroup of finite index, it is shown that $ D $ is a crossed product division algebra.