TITLE: Conceptual doubt about Modules
QUESTION [5 upvotes]: Let $R$ be a ring. Then a left $R$- module is an additive Abelian group M along with an operation 
$R\times M \to M$ such that it satisfies -

$r(x+y)=rx+ry$
$(r+s)x=rx+sx$
$(rs)x=r(sx)$
$1_R x=x$    

where $x,y \in M $ and $r \in R$
Now my question is why do we require a module to be an abelian group?. Can't we turn a non-abelian group into a module over some ring? Say if we have $G=S_3\ \text{or}\  GL_2(\Bbb{Z})$ and $R$ be some ring (commutative/noncommutative)

REPLY [5 votes]: Another perspective is the following. If $A$ is an abelian group, then the endomorphism monoid $\text{End}(A)$ naturally has the structure of a ring: multiplication is composition, and addition is pointwise addition. One way to describe what a (left) $R$-module structure on $A$ is is that it is a ring homomorphism $R \to \text{End}(A)$. (Compare: one way to describe what a group action on a set is is that it is a group homomorphism $G \to \text{Aut}(X)$.) 
By contrast, if $A$ is not necessarily abelian, then $\text{End}(A)$ only has the structure of a monoid: the problem is that the pointwise sum of two endomorphisms need not be an endomorphism, so there's no candidate for addition available.