Module

The category of modules over a fixed commutative ring $R$ are the prototypical abelian category; this statement is deeper than it may appear, in fact every small abelian category is equivalent to a full subcategory of some category of modules over a ring. This result is due to Freyd and Mitchell.

Definition

Let $R$ be a commutative ring with $1$. A (left) $R$-module consists of

1. An abelian group $M$
2. an action of $R$ on $M$; i.e., a map $R\times M\to M$, denoted by $(r,m)\mapsto r\cdot m$, such that
   1. $r\cdot (m_{1}+m_{2})=r\cdot m_{1}+r\cdot m_{2}$
   2. $(rr')\cdot m_{1}=r\cdot (r'\cdot m_{1})$
   3. $1\cdot m_{1}=m_{1}$

The category of $R$-modules

The morphisms in the category of $R$-modules are defined respecting the abelian group structure and the action of $R$. That is, a morphism $\varphi :M\to M'$ is a homomorphism $\varphi$ of the abelian groups $M$ and $M'$ such that $\varphi (r\cdot m)=r\cdot \varphi (m)$ for all $m\in M$.

Examples

1. The category of $\mathbb {Z}$-modules is equivalent to the category of abelian groups.