Module over a commutative unital ring

This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra

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.:

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

and:

$1.m=m\mathrm{\forall }m\in M$

• $.$ 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:

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

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