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
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 both cases for a left module and a right module.

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

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:
 * Left Module over Commutative Ring induces Right Module
 * Right Module over Commutative Ring induces Left Module.

But this is not the case for a ring that is not commutative. From:
 * Left Module Does Not Necessarily Induce Right Module over Ring
 * Leigh.Samphier/Sandbox/Right Module Does Not Necessarily Induce Left Module over Ring

it is known that it is not sufficient to simply reverse the scalar multiplication to get a module of the other ‘side’.

From:
 * Left Module over Ring Induces Right Module over Opposite Ring
 * Right Module over Ring Induces Left Module over Opposite Ring

to obtain a module of the other ‘side’ it is, in general, also necessary to reverse the product of the ring.

For this latter reason, all modules in a definition or theorem are necessarily assumed to be either all left modules or all right modules and never a mix of both.

In definitions and theorems involving a mix of left modules and right modules it is necessary to explicitly identify the left modules and right modules and nothing should be identified as simply a module.

On  the proof of a theorem involving modules  will be given for the left module case. The proof for the right module case will hold without being 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