Definition:Module

Definition
Let $\left({R, +_R, \times_R}\right)$ be a ring.

Let $\left({G, +_G}\right)$ be an abelian group.

A module over $R$ is an $R$-algebraic structure with one operation $\left({G, +_G, \circ}\right)_R$ 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 $\left({G, +_G}\right)$ 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 homomorphism to the endomorphism ring.

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

Also see

 * Definition:Scalar Ring

Special cases

 * Definition:Vector Space