Dashkova, O. (2015). Modules over group rings of groups with restrictions on the system of all proper subgroups. International Journal of Group Theory, 4(4), 43-48. doi: 10.22108/ijgt.2015.5504

Olga Dashkova. "Modules over group rings of groups with restrictions on the system of all proper subgroups". International Journal of Group Theory, 4, 4, 2015, 43-48. doi: 10.22108/ijgt.2015.5504

Dashkova, O. (2015). 'Modules over group rings of groups with restrictions on the system of all proper subgroups', International Journal of Group Theory, 4(4), pp. 43-48. doi: 10.22108/ijgt.2015.5504

Dashkova, O. Modules over group rings of groups with restrictions on the system of all proper subgroups. International Journal of Group Theory, 2015; 4(4): 43-48. doi: 10.22108/ijgt.2015.5504

Modules over group rings of groups with restrictions on the system of all proper subgroups

^{}Professor of the Branch of Moscow state university in Sevastopol

Abstract

We consider the class $\mathfrak M$ of $\bf R$--modules where $\bf R$ is an associative ring. Let $A$ be a module over a group ring $\bf R$$G$, $G$ be a group and let $\mathfrak L(G)$ be the set of all proper subgroups of $G$. We suppose that if $H \in \mathfrak L(G)$ then $A/C_{A}(H)$ belongs to $\mathfrak M$. We investigate an $\bf R$$G$--module $A$ such that $G \not = G'$, $C_{G}(A) = 1$. We study the cases: 1) $\mathfrak M$ is the class of all artinian $\bf R$--modules, $\bf R$ is either the ring of integers or the ring of $p$--adic integers; 2) $\mathfrak M$ is the class of all finite $\bf R$--modules, $\bf R$ is an associative ring; 3) $\mathfrak M$ is the class of all finite $\bf R$--modules, $\bf R$$=F$ is a finite field.

O. Yu. Dashkova (2013). Lo cally soluble AFA-groups. Ukr. Math. J., (in Russian). 65 (4), 459-469

O. Yu. Dashkova (2013). On linear groups with restrictions on the system of all
proper subgroups. Reports of the Academy of Sciences of Ukraine. (12), 7-10

O. Yu. Dashkova (2009). On modules over group rings of locally soluble groups for a ring of p-adic integers. Algebra Discrete Math. (1), 32-43

O. Yu. Dashkova (2013). On modules over group rings of groups with restrictions on the system of all proper subgroups, arXiv:1305.0744v2. , 6

M. R. Dixon, M. J. Evans, L. A. Kurdachenko (2004). Linear groups with the minimal condition on subgroups of infinite central dimension. J. Algebra. 277 (1), 172-186

L. Fuchs (1973). Infinite Abelian Groups. 1 Academic Press, New York.

P. M. Gudivok, V. P. Rudko, V. A. Bovdi (2006). Crystal lographic groups. Uzhgoro d: Uzhgoro dskii Natsionalnii Universitet,
(in Ukrainian). , 174

M.I. Kargap olov, Yu.I. Merzlyakov (1975). Bases of the Theory of Groups. Moscow, Nauka, (in Russian).

O. H. Kegel, B. A. F. Wehrfritz (1973). Locally Finite Groups. North-Holland Mathematical Library, North-Holland,
Amsterdam, London. 3

L. A. Kurdachenko, I. Ya. Subbotin (2006). Linear groups with the maximal condition on subgroups of infinite central dimension. Publ. Mat.. 50 (1), 103-131

L. A. Kurdachenko, I. Ya. Subbotin, N. N. Semko (2008). Insight into Modules over Dedekind Domains. National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev.

L. A. Kurdachenko, I. Ya. Subbotin, V. A. Chepurdya (2013). On the structure of some modules over generalized soluble
groups. arXiv:1302.2115. , 9

L. A. Kurdachenko, J. Otal, I. Ya. Subbotin (2007). Artinian Modules over Group Rings. Birkhauser, Boston, Berlin.

L. A. Kurdachenko (1993). On groups with minimax classes of conjugate elements. Infinite groups and adjoining algebraic structures, Academy of Sciences of Ukraine, Institute of Mathematics, Kiev,
(in Russian). , 160-177

A. G. Kurosh (1967). The Theory of Groups. Moscow, Nauka, (in Russian).

B. A. F. Wehrfritz (2005). Artinian-finitary groups are locally normal-finitary. J. Algebra. 287 (2), 417-431

B. A. F. Wehrfritz (2004). Artinian-finitary groups over commutative rings and non-commutative rings. J. Lond. Math. Soc. (2). 70 (2), 325-340

B. A. F. Wehrfritz (2003). Artinian-finitary groups over commutative rings. Illinois J. Math.. 47 (1-2), 551-565

B. A. F. Wehrfritz (2002). Finite-finitary groups of automorphisms. J. Algebra Appl.. 1 (4), 375-389

B. A. F. Wehrfritz (2002). On generalized finitary groups. J. Algebra. 247 (2), 707-727