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RESEARCH ARTICLE

On modules over group rings of locally soluble groups for a ring of $p$-adic integers

O. Yu. Dashkova

Department of Mathematics and Mechanics, Kyev National University, ul. ladimirskaya, 60, Kyev, 01033, Ukraine

Abstract: The author studies the ${\bf Z_{p^{\infty}}}G$-module $A$ such that $\bf Z_{p^{\infty}}$ is a ring of $p$-adic integers, a group $G$ is locally soluble, the quotient module $A/C_{A}(G)$ is not Artinian $\bf Z_{p^{\infty}}$-module, and the system of all subgroups $H \leq G$ for which the quotient\linebreak modules $A/C_{A}(H)$ are not Artinian $\bf Z_{p^{\infty}}$-modules satisfies the minimal condition on subgroups. It is proved that the group $G$ under consideration is soluble and some its properties are obtained.

Keywords: Linear group, Artinian module, locally soluble group.

MSC: 20F19, 20H25

Received: 22.03.2009
Revised: 30.04.2009

Language: English

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