Definition:Module

Definition
Let $\left({R, +_R, \times_R}\right)$ be a ring.

Let $\left({G, +_G}\right)$ be an abelian group.

Module Axioms
Note that a module is not an algebraic structure unless $R$ and $G$ are the same set.

Vector
The elements of $\left({G, +_G}\right)$ are called vectors.

Also known as
A module over $R$ can also be referred to as an $R$-module'''.

Also see

 * Definition:Scalar Ring


 * Definition:Vector Space