Do we have a notion of $G-$module where $G$ is only a group without a ring structure?

Question

Do we have a notion of $G-$module where $G$ is only a group without a ring structure ? If yes what do we call it and where do we use it. but if not why we are interested in $R-$modules where $R$ is a ring and not in $G-$modules where $G$ is only a group?

Answer 1

Yes, called a $G$-module as you expected!

The category $\bf{G-Mod}$ of (left) $G$ modules can be identified with the category of (left) $Z[G]$-modules, i.e. with the modules over the group ring $Z[G]$, so this is secretly still an $R$-module for a ring $R$. Maybe this explains why these aren't talked about so often.