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
References: [1] R. Brauer, W. Feit: An analogue of Jordan's theorem in characteristic $p$. Ann. Math. (2) 84 (1966), 119-131. DOI 10.2307/1970514 | MR 0200350 | Zbl 0142.26203 [2] M. R. Dixon, L. A. Kurdachenko, J. Otal: Linear analogues of theorems of Schur, Baer and Hall. Int. J. Group Theory 2 (2013), 79-89. MR 3033535 | Zbl 1306.20055 [3] L. A. Kurdachenko, I. Ya. Subbotin, V. A. Chupordia: On the relations between the central factor-module and the derived submodule in modules over group rings. Commentat. Math. Univ. Carol. 56 (2015), 433-445. DOI 10.14712/1213-7243.2015.136 | MR 3434223 | Zbl 1345.20008 [4] J. C. McConnell, J. C. Robson: Noncommutative Noetherian Rings. With the Cooperation of L. W. Small. Pure and Applied Mathematics. A Wiley-Interscience Publication, John Wiley & Sons, Chichester (1987). MR 0934572 | Zbl 0644.16008 [5] B. A. F. Wehrfritz: Infinite Linear Groups. An Account of the Group-Theoretic Properties of Infinite Groups of Matrices. Ergebnisse der Mathematik und ihrer Grenzgebiete 76, Springer, Berlin (1973). DOI 10.1007/978-3-642-87081-1 | MR 0335656 | Zbl 0261.20038 [6] B. A. F. Wehrfritz: Automorphism groups of Noetherian modules over commutative rings. Arch. Math. 27 (1976), 276-281. DOI 10.1007/BF01224671 | MR 0409615 | Zbl 0333.13009 [7] B. A. F. Wehrfritz: On the Lie-Kolchin-Mal'cev theorem. J. Aust. Math. Soc., Ser. A 26 (1978), 270-276. DOI 10.1017/S1446788700011782 | MR 0515743 | Zbl 0392.20026 [8] B. A. F. Wehrfritz: Lectures around Complete Local Rings. Queen Mary College Mathematics Notes, London (1979). MR 0550883 [9] B. A. F. Wehrfritz: Group and Ring Theoretic Properties of Polycyclic Groups. Algebra and Applications 10, Springer, Dordrecht (2009). DOI 10.1007/978-1-84882-941-1 | MR 2561933 | Zbl 1206.20042
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
