On the structure of artinian-by-(finite rank) modules over generalized soluble groups

V.A. Chupordia (Oles Honchar Dnipropetrovsk National University)

Abstract


Let R be a ring and G a group. An R-module A is said to be artinian-by-(finite rank), if $$$\mathrm{Tor}_R(A)$$$ is artinian and $$$A/\mathrm{Tor}_R(A)$$$ has finite R-rank. In this paper modules A over a group ring $$${\mathbb{Z}}_{p^{\infty}}G$$$ such that $$$A/C_A(H)$$$ is artinian-by-(finite rank) (as an $$${\mathbb{Z}}_{p^{\infty}}$$$-module) for every proper subgroup H are investigated.

Keywords


modules; group rings; modules over group rings; generalized soluble groups; radical groups; artinian modules; generalized radical groups; modules of finite rank

Full Text:

PDF

References


Charin V.S. "On the automorphism groups of nilpotent groups", Ukrain. Math. J., 1954; 6(3): pp. 295-304. (in Russian)

Dixon M.R., Evans M.J., Smith H. "Groups with all proper subgroups soluble-by-finite rank", Journal of Algebra, 2005; 289: pp. 135-147.

Fuchs L. Infinite abelian groups, Academic Press, New York, 1970; Vol. 1.

Kaloujnine L.A. "Uber gewisse Beziehungen zwischen eine Gruppe und ihren Automorphismen", Bericht Math Tagung Berlin, 1953; pp. 164-172 (in German)

Kurdachenko L.A., Otal J., Subbotin I.Ya. Artinian modules over group rings, Birkhauser, Basel, 2007.

Kurdachenko L.A., Subbotin I.Ya., Chupordya V.A. "On bounded artinian finitary modules", International Journal of Algebra and Computation, 2007; 17(4): pp. 881-893.

Kurdachenko L.A., Subbotin I.Ya., Chupordya V.A. "On the structure of some modules over generalized soluble groups", Turkish Journal of Mathematics, 2014; 38: pp. 52-59.

Napolitani F., Pegoraro E. "On groups with nilpotent by Chernikov proper subgroup", Arch. Math., 1997; 69: pp. 89-94.

Phillips R. "Finitary linear groups: a survey", Finite and locally finite groups, NATO ASI ser C 471 Dordrecht, Kluwer 1995; pp. 111-146.

Wehrfritz B.A.F. Infinite linear groups, Springer, Berlin, 1973.

Wehrfritz B.A.F. "Finite-finitary groups of automorphisms", Journal of Algebra and Its Applications, 2002; 4: pp. 375-389.

Wehrfritz B.A.F. "On generalized finitary groups", Journal of Algebra, 2002; 247: pp. 707-727.

Wehrfritz B.A.F. "Finitary automorphism groups over commutative rings", Journal Pure Appl. Algebra, 2002; 172: pp. 337-346.

Wehrfritz B.A.F. "Finitary and artinian-finitary groups over the integers $$$\mathbb{Z}$$$", Ukrainian Math. Journal, 2002; 54: pp. 924-936.

Wehrfritz B.A.F. "Artinian-finitary groups over commutative rings", Illinois Journal of Math., 2003; 47: pp. 551-565.

Wehrfritz B.A.F. "Finitary and artinian-finitary groups over commutative rings", Journal Group Theory, 2004; 7: pp. 243-253.

Wehrfritz B.A.F. "Artinian-finitary groups over commutative rings and non-commutative rings", Journal London Math. Soc., 2004; 70: pp. 325-340.

Wehrfritz B.A.F. "Artinian-finitary groups are locally normal-finitary", Journal of Algebra, 2005; 287: pp. 417-431.




DOI: https://dx.doi.org/10.15421/241412

  

Refbacks

  • There are currently no refbacks.


Copyright (c) 2014 V.A. Chupordia

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Registered in


ISSN (Online): 2664-5009
ISSN (Print): 2664-4991
DNU