Representations of certain groups with finiteness conditions and modules over group rings

We consider modules over group rings and representations of the following groups: soluble-by-finite groups of finite rank, linear groups of finite rank, locally normal groups, some locally finite groups. We search for conditions when these groups have faithful irreducible representations. Induced representations of linear groups of finite rank are studied. We also study ideals of group rings  of abelian groups of finite rank and modules over these group rings.