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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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Lenguaje: | eng |
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2002
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Acceso en línea: | http://cds.cern.ch/record/645825 |
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author | Kuku, A O Mahdavi-Hezavehi, M |
author_facet | Kuku, A O Mahdavi-Hezavehi, M |
author_sort | Kuku, A O |
collection | CERN |
description | 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. |
id | cern-645825 |
institution | Organización Europea para la Investigación Nuclear |
language | eng |
publishDate | 2002 |
record_format | invenio |
spelling | cern-6458252019-09-30T06:29:59Zhttp://cds.cern.ch/record/645825engKuku, A OMahdavi-Hezavehi, MSubgroups of $GL_{n}(R)$ for local rings $R$XXLet $ 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.IC-2002-66oai:cds.cern.ch:6458252002 |
spellingShingle | XX Kuku, A O Mahdavi-Hezavehi, M Subgroups of $GL_{n}(R)$ for local rings $R$ |
title | Subgroups of $GL_{n}(R)$ for local rings $R$ |
title_full | Subgroups of $GL_{n}(R)$ for local rings $R$ |
title_fullStr | Subgroups of $GL_{n}(R)$ for local rings $R$ |
title_full_unstemmed | Subgroups of $GL_{n}(R)$ for local rings $R$ |
title_short | Subgroups of $GL_{n}(R)$ for local rings $R$ |
title_sort | subgroups of $gl_{n}(r)$ for local rings $r$ |
topic | XX |
url | http://cds.cern.ch/record/645825 |
work_keys_str_mv | AT kukuao subgroupsofglnrforlocalringsr AT mahdavihezavehim subgroupsofglnrforlocalringsr |