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MATHEMATICS

Locally soluble $\operatorname{AFN}$-groups

O. Yu. Dashkova

O. Honchar Dnepropetrovsk National University, Dnepropetrovsk, Ukraine

Abstract: Let $A$ be an $\textrm{R}G$-module, where $\textrm{R}$ is a commutative noetherian ring with the unit, $G$ is a locally soluble group, $C_G(A) = 1$, and each proper subgroup $H$ of a group $G$ for which $A/C_A(H)$ is not a noetherian $\textrm{R}$-module, is finitely generated. It is proved that a locally soluble group $G$ with these conditions is hyperabelian. It is described the structure of a group $G$ under consideration if $G$ is a finitely generated soluble group and the quotient module $A/C_A(G)$ is not a noetherian $\textrm{R}$-module.

Keywords: group ring, locally soluble group, noetherian $\textrm{R}$-module.

UDC: 512.544

Received: 09.02.2012