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: b.a.f.wehrfritz@qmul.ac.uk

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.


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


Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer need to access the articles on their site, which is https://link.springer.com/journal/10587.

[Previous Article] [Next Article] [Contents of This Number] [Contents of Czechoslovak Mathematical Journal]