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.
Similar content being viewed by others
References
R. Brauer, W. Feit: An analogue of Jordan’s theorem in characteristic p. Ann. Math. 2 (84) (1966), 119–131.
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.
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.
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.
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.
B. A. F. Wehrfritz: Automorphism groups of Noetherian modules over commutative rings. Arch. Math. 27 (1976), 276–281.
B. A. F. Wehrfritz: On the Lie-Kolchin-Mal’cev theorem. J. Aust. Math. Soc., Ser. A 26 (1978), 270–276.
B. A. F. Wehrfritz: Lectures around Complete Local Rings. Queen Mary College Mathematics Notes, London, 1979.
B. A. F. Wehrfritz: Group and Ring Theoretic Properties of Polycyclic Groups. Algebra and Applications 10, Springer, Dordrecht, 2009.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wehrfritz, B.A.F. On soluble groups of module automorphisms of finite rank. Czech Math J 67, 809–818 (2017). https://doi.org/10.21136/CMJ.2017.0193-16
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2017.0193-16