Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 809-818, 2017


On soluble groups of module automorphisms of finite rank

Bertram A. F. Wehrfritz

Received April 18, 2016.   First published August 9, 2017.

Abstract:  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.
Keywords:  soluble group; finite rank; module automorphisms; Noetherian module over commutative ring
Classification MSC:  20F16, 20C07, 13E05, 20H99
DOI:  10.21136/CMJ.2017.0193-16


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Affiliations:   Bertram A. F. Wehrfritz, School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, England, e-mail: b.a.f.wehrfritz@qmul.ac.uk

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