Module

In abstract algebra, a module is a mathematical structure of which abelian groups and vector spaces are particular types. They have become ubiquitous in abstract algebra and other areas of mathematics that involve algebraic structures, such as algebraic topology, algebraic geometry, and algebraic number theory. A strong understanding of module theory is essential for anyone desiring to understand a wide array of graduate level mathematics and current mathematical research.

Definition

Let $R$ be a ring (not necessarily with identity or commutative). A left $R$-module is an abelian group whose underlying set is endowed with an action (mathematics) by $R$ respecting both the group structure of $M$ and the ring structure of $R$. The $R$ action is a map $R\times M\rightarrow M$. The image of $r\times m$ under this map is typically written $r\cdot m$, or just $rm$. The action is required to satisfy the following properties:

1. $r\cdot (m_{1}+m_{2})=r\cdot m_{1}+r\cdot m_{2}$, for all $r\in R,m_{1},m_{2}\in M$
2. $(r_{1}+r_{2})\cdot m=r_{1}\cdot m+r_{2}\cdot m$, and
3. $(r_{1}r_{2})\cdot m=r_{1}\cdot (r_{2}\cdot m)$ for all $r_{1},r_{2}\in R,m\in M$

If the ring $R$ has an identity, a module satisfying the additional axiom

$1\cdot m=m$ for all $m\in M$

is called unital or unitary.

A right $R$-module can be defined similarly.

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 $m\in M$.

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.

Examples

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