Module

A module is a generalization of a vector space but the scalars are from a ring.

Suppose that $R$ is a ring with multiplicative identity 1. A left $R$-module $M$ is an abelian group $(M,+)$ with operation $\cdot :R\times M\to M$ such that

1. $r\cdot (x+y)=r\cdot x+r\cdot y$
2. $(r+s)\cdot x=r\cdot x+s\cdot x$
3. $(rs)\cdot x=r\cdot (s\cdot x)$
4. $1\cdot x=x$

A right $R$-module is defined similarly but we right-multiply by the scalar.