## Left module

### Set

 context $M,R$
 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.

 $x,y\in M$ $r,s\in R$
 postulate $r*(x+y) = (r*x)+(r*y)$ postulate $(r\ \hat+\ s)* x = (r* x)+(s* x)$ postulate $(r\ \hat*\ s)* x = r* (s* x)$

### Discussion

“$*$” 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.

#### Reference

Wikipedia: Module