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.

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