R-module

Definition

Let $R$ be a ring. A (left) $R$-module is an action of $R$ on an abelian group $M$. More explicitly, this means that an $R$-module $M$, (equivalently a left action of a ring $R$ on $M$) is a homomorphism of rings

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

Equivalent definition

Even more explicitly, any homomorphism of rings $\sigma$ as defined above will have the following properties. This can be taken as the definition as well.

Let

$$\rho :R\times M⟶M$$

satisfying the requirements:

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

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

• $\rho (rs,m)=\rho (r,\rho (s,m))$

• $\rho (1,m)=m$

Examples

• A $\mathbb{Z}$-module is just an abelian group

• If $R=k$ for some field, then the $R$-modules are vector spaces

• Any homomorphism of rings $\alpha :R\to S$ defines an $R$ action by taking

$$\rho (r,s):=\alpha (r)s\forall r\in R,s\in S$$

NOTE

This is an important example because it is how we build R-algebras

todo ADD GROUP REPRESENTATIONS - and group ring

Morphisms

A homomorphism of $R$-modules $\phi :M\to N$ is a homomorphism of abelian groups such that $\forall m,\tilde{m}\in M,r\in R$,

• $\phi (m+\tilde{m})=\phi (m)+\phi (\tilde{m})$

• $\phi (rm)=r\phi (m)$

Quotient modules

Given $N\subset M$ a submodule of $R$, we define the quotient module $M/N$ in the same way as abelian groups (with extra stuff):

As a set:

$$M/N=\{m+N|m\in M\}$$

that is, $[m]=[p]⟺m-p\in N$.

The $R$-action: Since we know there exists a canonical projection map

$$\begin{array}{rl}\pi :M& ⟶N\\ m& ⟼[m]\end{array}$$

then we define

$$r\cdot (m+N)=rm+N$$

therefore, $\pi$ is a $R$-module homomorphism.