Difference between $\mathbb{Z} G$-module and $G$-module

Question

I am studying Group Cohomology, but I am a little confused about $\mathbb{Z} G$-module and $G$-module. Some text uses the $G$-module for group cohomology, but I thought group cohomology of $G$ is defined on $\mathbb{Z} G$-module. Are they isomorphic in some way? Why could we use only $G$-module instead of $\mathbb{Z} G$-module?

Answer 1

$G$-modules, defined as abelian groups with an associative unital distributive action of $G$, are the same thing as $\mathbb{Z}G$-modules. For a $\mathbb{Z}G$-module restricts to a $G$-module via the inclusion of $G$ into the group ring, a $G$-module induces a $\mathbb{Z}G$-module by linearity, and these operations are mutually inverse.