| context | $M,R$ |
| postulate | $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{module}(\mathcal M,\mathcal R)$ |
| context | $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(\mathcal M,\mathcal R)$ |
| context | $\mathcal M\in \mathrm{abelianGroup}(M)$ |
Now denote the multiplication in the ring $\mathcal R$ by “$\ \hat*\ $”.
| $r,s\in R$ |
| postulate | $r*s=s*r$ |
A module is a left module with a commutative ring acting on the group.
One generally speaks of an $R$-module over $M$. Here $R$ and $M$ are just sets.
Wikipedia: Module