I am trying to understand the equivalence between group representations, $(V, \rho)$, and left modules over the group ring $F[G]$. Can you explain explicitly why it is the same?
My progress: Consider a group $G$. If $(V, \rho)$ is a representation of $G$, we can take $V$ to be a left module over $F[G]$ by defining: $gv = \rho(g)v$. So given a representation we can get a left module over $F[G]$.
The other direction is more confusing. Given a left module $M$ over $F[G]$, what is the vector space? Is $M$ necessarily a vector space? (The concept of a module is rather new to me).