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 either a left module or a right module, the type is unspecified:

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

Note
A theorem about modules, where the type of modules is unspecified, is in effect two theorems: one about left modules where all modules are left modules; the other about right modules where all modules are right modules

The theorem is never about a mixture of left and right modules.

The proof of such a theorem is generally given for one type of module only. The proof for the other type is proved similarly with the scalar applied on the other side.

For example, see Direct Product of Modules is Module.

Where a theorem does involve a mixture of left and right modules it is necessary to explicitly identify which modules are the left modules and which are the right modules.

For example, see Left Module over Ring Induces Right Module over Opposite Ring.

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