If V is an FG-module according to your definition, then V can easily be made into a module over the ring FG. That is, consider V as an abelian group, and let the scalars from the ring FG act on V. You have an action of the group G on V; extend this linearly to get an action of the ring FG on V. As the name suggests, an FG-module is just a module over the group ring FG.
Yes and the reason why we look at the FG-module as a module rather than a vectorspace is because decomposing it into a sum of irreducible submodules corresponds to decomposing the according representation into its irreducible parts. If we just consider it as a vectorspace we don't get this since then we just have a direct sum of 1-dimensional subspaces (the ones spanned by the basis elements)
If V is an FG-module according to your definition, then V can easily be made into a module over the ring FG. That is, consider V as an abelian group, and let the scalars from the ring FG act on V. You have an action of the group G on V; extend this linearly to get an action of the ring FG on V. As the name suggests, an FG-module is just a module over the group ring FG.
Yes and the reason why we look at the FG-module as a module rather than a vectorspace is because decomposing it into a sum of irreducible submodules corresponds to decomposing the according representation into its irreducible parts. If we just consider it as a vectorspace we don't get this since then we just have a direct sum of 1-dimensional subspaces (the ones spanned by the basis elements)
Very nice (and a bit fast)--thank you so much!