Conceptual doubt about Modules

Question

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)

Answer 1

Commutativity of the operation of $M$ is forced from the other axioms. Indeed, if $x,y\in M$, then $$ 0 = 0_R(x+y) = (1_R-1_R)(x+y) = 1_R(x+y) - 1_R(x+y) = x+y-x-y. $$

Thus, adding $y$, then $x$ on the right, we get $y+x=x+y$.

Answer 2

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.