Definition:Module/Left and Right Modules

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 either a left module or 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.

Note
In the case of a commutative ring, the difference between a left module and a right module is little more than a notational difference. See

A module over $R$ is an $R$-algebraic structure where the ‘side’ the actions are applied is unspecified. This is a convenience that allows definitions and theorems to be stated without the tedious duplication of the the cases for a left module and a right module.

This means that an definitions and theorems involving modules over a ring $R$ are equally valid for left modules as for right modules.

Whats more, where the ‘side’ the actions is unspecified all modules are necessarily assumed to have actions on the same ‘side’.

On the other hand, discussions involving module with actions on different ‘sides’ must have the ‘sides’ explicitly stated.

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