Module over a commutative unital ring

Definition
Let $$R$$ be a commutative unital ring. A module over $$R$$ is an Abelian group $$M$$ along with a map $$.: R \times M \to M$$ such that:


 * $$.$$ is a monoid action of the multiplicative monoid of $$R$$ on $$M$$, viz.:

$$a.(b.m) = (ab).m \ \forall \ a,b \in R, \ m \in M$$

and:

$$1.m = m \ \forall \ m \in M$$


 * $$.$$ is an additive homomorphism from $$R$$ (treated as an additive group) to the additive group of all functions from $$M$$ to itself, under pointwise addition. In symbols:

$$(a + b).m = a.m + b.m \ \forall \ a,b \in R, \ m \in M$$

It follows that $$0.m = 0$$ and $$(-a).m = -(a.m)$$


 * The map $$m \mapsto a.m$$ is an endomorphism of $$M$$, viewed as an Abelian group.

All the above three conditions can be stated concisely as: the map $$R \times M \to M$$ homomorphism of unital rings $$f:R \to End(M)$$, where $$r.m := f(r)(m)$$.