| definiendum | $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(M,R)$ |
| context | $\mathcal M\in \mathrm{abelianGroup}(M)$ |
| context | $\mathcal R\in \mathrm{ring}(R)$ |
| context | $*:R\times M\to M$ |
Now denote the addition in th group $\mathcal M$ by “$+$” as usual, and the addition and multiplication in the ring $\mathcal R$ by “$\hat+$” and “$\hat*$”, respectively.
| postulate | $r*(x+y) = (r*x)+(r*y)$ |
| postulate | $(r\ \hat+\ s)* x = (r* x)+(s* x)$ |
| postulate | $(r\ \hat*\ s)* x = r* (s* x)$ |
“$*$” is an action of the ring on the group from the left. If the ring is commutative, then one need not distinguish between left- and right module.
One generally speaks of an $R$-left-module over $M$. Here $R$ and $M$ are just sets.