On soluble groups of module automorphisms of finite rank

Let R be a commutative ring, M an R -module and G a group of R -automorphisms of M , usually with some sort of rank restriction on G . We study the transfer of hypotheses between M/C M ( G ) and [ M,G ] such as Noetherian or having finite composition length. In this we extend recent work of Dixon, K...

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Bibliographic Details
Published inCzechoslovak mathematical journal Vol. 67; no. 3; pp. 809 - 818
Main Author Wehrfritz, Bertram A. F.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2017
Springer Nature B.V
Subjects
Analysis
Automorphisms
Convex and Discrete Geometry
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
13E05
soluble group
module automorphisms
Noetherian module over commutative ring
20F16
finite rank
20C07
20H99
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Summary:Let R be a commutative ring, M an R -module and G a group of R -automorphisms of M , usually with some sort of rank restriction on G . We study the transfer of hypotheses between M/C M ( G ) and [ M,G ] such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose [ M,G ] is R -Noetherian. If G has finite rank, then M/C M ( G ) also is R -Noetherian. Further, if [ M,G ] is R -Noetherian and if only certain abelian sections of G have finite rank, then G has finite rank and is soluble-by-finite. If M/C M ( G ) is R -Noetherian and G has finite rank, then [ M,G ] need not be R -Noetherian.
ISSN:0011-4642
1572-9141
DOI:10.21136/CMJ.2017.0193-16