# Module over a commutative unital ring

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## 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 $r.m := f(r)(m)$.