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

We consider the class m of r-modules where r is an associative ring. let a be a module over a group ring rg, g be a group and let l(g) be the set of all proper subgroups of g. we suppose that if h ∈ l(g) then a/ca(h) belongs to m. we study an rg-module a such that g ≠ g', cg(a) = 1. we consider the cases: 1) m is the class of all artinian r-modules, r is either the ring of integers or the ring of p-adic integers; 2) m is the class of all finite r-modules, r is an associative ring; 3) m is the class of all finite r-modules, r= f is a finite field.