It is traditional to say that a representation of a group $G$ over a field $F$ is "exactly like" a module over the group ring $F[G]$.
I think it is inaccurate. I think a module over $F[G]$ encodes more than a representation of $G$. I will give an example of two modules over $F[G]$ giving rise to the same group representation of $G$.
I will give the simplest example:
Let $G$ be the trivial group of order $1$, and let $F=\mathbb{C}$. I can give the abelian group $M=\mathbb{C}$ a structure of a $\mathbb{C}[G]$-module in different ways. I can let each $z\cdot 1_G\in\mathbb{C}$ act on $m\in M$ either by $(z\cdot 1_G)\cdot m=zm$, or by $(z\cdot 1_G)\cdot m=\overline{z}m$.
Is it true to say that there is a bijective correspondence between the following collections?
- The collection of pairs $(\rho,\theta)$, where $\rho$ is a group representation of $G$ over $F$, and $\theta$ is a field endomorphism of $F$, and
- The collection of modules over $F[G]$