I am reading a book in representation theory by James and Liebeck. They define an FG-module as: Let $V$ be a vector space over the field $F$ and let $G$ be a group. Then $V$ is an FG-module if a multiplication $vg$ $(v \in V, g \in G)$ is defined, satisfying the following conditions for all $ u,v \in V, \lambda \in F $ and $g,h \in G $:
(1) $vg \in V$, (2) $ v(gh) = (vg)h$, (3) $v1=v$, (4) $(\lambda v)g = \lambda(vg)$, (5) $(u+v)g =ug + vg$
They state that they use the letters $F$ and $G$ to indicate that $V$ is a vector space over $F$ and that the elements $g$ comes from $G$. I am asking for analogs to the definitions of a R-module, or any discussion, clarification or generalisation of the definition above. Also if $V$ is an FG-module as defined above then there is a basis for the module and hence it is finitely generated? In the following chapters the connections between FG-modules and group representations are explored, but I wonder if some of these results may hold for any finite generated module, what is the connection? Also I wonder if there is any representation theoretical approach to prove the Structure theorem for finitely generated modules over a principal ideal domain or any special case of this theorem, because I suspect there is... I am a totally beginner to representation theory and modules and group algebras so I am basically looking for a bigger picture to give things a little bit more connection and motivation.