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.

Bertram A. F. Wehrfritz, School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, England, e-mail:

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 2010): 20F16, 20C07, 13E05, 20H99

DOI: 10.21136/CMJ.2017.0193-16

Full text available as PDF.

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