module over ring

Definition

For a ring $R$, a (left) $R$-module is a left-action of a ring $R$ on an abelian group $M$. This is a homomorphism of rings

$$\sigma :R⟶{\text{End}}_{\text{Ab}}(M).$$

Right $R$-modules can be defined analogously.

For a commutative ring, the right and left notion is not needed.

Alternate definition

It can be proved that the above definition is equivalent to the existence of a function

$$\rho :R\times M⟶M$$

satisfying the requirements below:

• $\rho (r,m+n)=\rho (r,m)+\rho (r,n)$

• $\rho (r+s,m)=\rho (r,m)+\rho (s,m)$

• $\rho (rs,m)=\rho (r,\rho (s,m))$

• $\rho (1,m)=m$

Notation

Often the $\rho$ in the definition above is dropped and $\rho (r,m)$ is denoted $rm$.

As a category

The modules over a ring $R$ make a category denoted $R\text{-Mod}$.

Objects: $R$-modules, that is abelian groups $M$ along with the homomorphism $R\to {\text{End}}_{\text{Ab}}(M)$ as defined above.

Morphisms: A homomorphism of $R$-modules is a group homomorphism $\phi :M\to N$ such that

• $\phi (m_1+m_2)=\phi (m_1)+\phi (m_2)$ for all $m_1,m_2\in M$

• $\phi (rm)=r\phi (m)$ for all $m\in M,r\in R$.

Composition is defined as composition of group homomorphisms.

Submodules

A submodule $N\subset M$ (of a $R$-module $M$) is a subgroup that is preserved by the action of $R$. That is,

$$rn\in N\forall r\in R,n\in N$$

In this way, $N$ is an $R$-module itself and the inclusion

$$N\hookrightarrow M$$

is a $R$-module homomorphism.

Examples

If $R=k$ for some field $k$, then an $R$-module is a vector space (which defines it’s own category).

Every abelian group is a $\mathbb{Z}$-module.