| Title:
|
On soluble groups of module automorphisms of finite rank (English) |
| Author:
|
Wehrfritz, Bertram A. F. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
3 |
| Year:
|
2017 |
| Pages:
|
809-818 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
soluble group |
| Keyword:
|
finite rank |
| Keyword:
|
module automorphisms |
| Keyword:
|
Noetherian module over commutative ring |
| MSC:
|
13E05 |
| MSC:
|
20C07 |
| MSC:
|
20F16 |
| MSC:
|
20H99 |
| idZBL:
|
Zbl 06770132 |
| idMR:
|
MR3697918 |
| DOI:
|
10.21136/CMJ.2017.0193-16 |
| . |
| Date available:
|
2017-09-01T12:25:49Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146861 |
| . |
| Reference:
|
[1] Brauer, R., Feit, W.: An analogue of Jordan's theorem in characteristic $p$.Ann. Math. (2) 84 (1966), 119-131. Zbl 0142.26203, MR 0200350, 10.2307/1970514 |
| Reference:
|
[2] Dixon, M. R., Kurdachenko, L. A., Otal, J.: Linear analogues of theorems of Schur, Baer and Hall.Int. J. Group Theory 2 (2013), 79-89. Zbl 1306.20055, MR 3033535 |
| Reference:
|
[3] Kurdachenko, L. A., Subbotin, I. Ya., Chupordia, V. A.: 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. Zbl 1345.20008, MR 3434223, 10.14712/1213-7243.2015.136 |
| Reference:
|
[4] McConnell, J. C., Robson, J. C.: Noncommutative Noetherian Rings. With the Cooperation of L. W. Small.Pure and Applied Mathematics. A Wiley-Interscience Publication, John Wiley & Sons, Chichester (1987). Zbl 0644.16008, MR 0934572 |
| Reference:
|
[5] Wehrfritz, B. A. F.: 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). Zbl 0261.20038, MR 0335656, 10.1007/978-3-642-87081-1 |
| Reference:
|
[6] Wehrfritz, B. A. F.: Automorphism groups of Noetherian modules over commutative rings.Arch. Math. 27 (1976), 276-281. Zbl 0333.13009, MR 0409615, 10.1007/BF01224671 |
| Reference:
|
[7] Wehrfritz, B. A. F.: On the Lie-Kolchin-Mal'cev theorem.J. Aust. Math. Soc., Ser. A 26 (1978), 270-276. Zbl 0392.20026, MR 0515743, 10.1017/S1446788700011782 |
| Reference:
|
[8] Wehrfritz, B. A. F.: Lectures around Complete Local Rings.Queen Mary College Mathematics Notes, London (1979). MR 0550883 |
| Reference:
|
[9] Wehrfritz, B. A. F.: Group and Ring Theoretic Properties of Polycyclic Groups.Algebra and Applications 10, Springer, Dordrecht (2009). Zbl 1206.20042, MR 2561933, 10.1007/978-1-84882-941-1 |
| . |
