Modules

Definition: Modules

Given a ring $R$, an $R$-module is an abelian group $M$ with binary operation $+$ and an action of $R$ on $M$. We denote the action of an element $r\in R$ on $m\in M$ as $rm$. An $R$-module then satisfies the following properties for all $r,r_1,r_2\in R$ and for all $m,m_1,m_2\in M$,

• $r_1(r_{2}m)=(r_1{r}_2)m$
• $r(m_1+m_2)=r{m}_1+r{m}_2$
• $(r_1+r_2)m=r_{1}m+r_{2}m$
• $1m=m$ (assuming $R$ has unity)

Since $R$ acts on the left, we call $M$ a left $R$-module.

We can think of modules as being a generalization of vector spaces where the elements of $R$ represent scalars and the elements of $M$ represent vectors.

Example Fest! §

There are four big examples of modules:

• Z-modules
• k-modules (field modules)
• [[1.6 Modules over Polynomial Rings|k[x]-modules (polynomial ring modules)]]
• [[k[G]-modules (group-algebras)]]