On soluble groups of module automorphisms of finite rank

Bertram A. F. Wehrfritz

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 3, page 809-818
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Wehrfritz, Bertram A. F.. "On soluble groups of module automorphisms of finite rank." Czechoslovak Mathematical Journal 67.3 (2017): 809-818. <http://eudml.org/doc/294104>.

@article{Wehrfritz2017,
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.},
author = {Wehrfritz, Bertram A. F.},
journal = {Czechoslovak Mathematical Journal},
keywords = {soluble group; finite rank; module automorphisms; Noetherian module over commutative ring},
language = {eng},
number = {3},
pages = {809-818},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On soluble groups of module automorphisms of finite rank},
url = {http://eudml.org/doc/294104},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Wehrfritz, Bertram A. F.
TI - On soluble groups of module automorphisms of finite rank
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 809
EP - 818
AB - 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.
LA - eng
KW - soluble group; finite rank; module automorphisms; Noetherian module over commutative ring
UR - http://eudml.org/doc/294104
ER -

References

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  3. Kurdachenko, L. A., Subbotin, I. Ya., Chupordia, V. A., 10.14712/1213-7243.2015.136, Commentat. Math. Univ. Carol. 56 (2015), 433-445. (2015) Zbl1345.20008MR3434223DOI10.14712/1213-7243.2015.136
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  5. Wehrfritz, B. A. F., 10.1007/978-3-642-87081-1, Ergebnisse der Mathematik und ihrer Grenzgebiete 76, Springer, Berlin (1973). (1973) Zbl0261.20038MR0335656DOI10.1007/978-3-642-87081-1
  6. Wehrfritz, B. A. F., 10.1007/BF01224671, Arch. Math. 27 (1976), 276-281. (1976) Zbl0333.13009MR0409615DOI10.1007/BF01224671
  7. Wehrfritz, B. A. F., 10.1017/S1446788700011782, J. Aust. Math. Soc., Ser. A 26 (1978), 270-276. (1978) Zbl0392.20026MR0515743DOI10.1017/S1446788700011782
  8. Wehrfritz, B. A. F., Lectures around Complete Local Rings, Queen Mary College Mathematics Notes, London (1979). (1979) MR0550883
  9. Wehrfritz, B. A. F., 10.1007/978-1-84882-941-1, Algebra and Applications 10, Springer, Dordrecht (2009). (2009) Zbl1206.20042MR2561933DOI10.1007/978-1-84882-941-1

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