Abstract:
Let $A$ be an $\mathrm{R}G$-module, where $\mathbf{R}$ is a commutative ring with the unit, $A/C_A(G)$ is not an artinian $\mathbf{R}$-module, $C_G(A) = 1$ and $G$ is a locally finite group. Let $\mathfrak{L}_{nad}(G)$ be a system of all subgroups $H \le G$ such that quotient modules $A/C_A(H)$ are not artinian $\mathbf{R}$-modules. The author studies $\mathbf{R}G$-module $A$ such that $\mathfrak{L}_{nad}(G)$ satisfies either weak minimal condition or weak maximal condition as an ordered set. The properties of the locally finite group $G$ with these conditions are described. Some properties of a locally soluble periodic group $G$ under consideration are obtained if $\mathbf{R}$ is a dedekind ring.
Keywords:an artinian $\mathbf{R}$-module, a group ring, a locally finite group.
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