Definition:Module

Definition
Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.

A module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$ which is both a left module and a right module:

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

Vector
The elements of $\struct {G, +_G}$ are called vectors.

Also defined as
Sources who only deal with rings with unity often define a module as what on is called a unitary module.

Sometimes no distinction is made between the module and the associated ring representation.

The word module can also be seen in some older works to mean vector magnitude or vector length.

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

Also see

 * Definition:Scalar Ring
 * Basic Results about Modules

Special cases

 * Definition:Unitary Module


 * Definition:Vector Space