# 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:

### Left Module

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

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

A left module over $R$ is an $R$-algebraic structure $\left({G, +_G, \circ}\right)_R$ with one operation $\circ$, the (left) ring action, which satisfies the left module axioms:

 $(1)$ $:$ $\displaystyle \forall \lambda \in R: \forall x, y \in G:$ $\displaystyle \lambda \circ \left({x +_G y}\right) = \left({\lambda \circ x}\right) +_G \left({\lambda \circ y}\right)$ $(2)$ $:$ $\displaystyle \forall \lambda, \mu \in R: \forall x \in G:$ $\displaystyle \left({\lambda +_R \mu}\right) \circ x = \left({\lambda \circ x}\right) +_G \left({\mu \circ x}\right)$ $(3)$ $:$ $\displaystyle \forall \lambda, \mu \in R: \forall x \in G:$ $\displaystyle \left({\lambda \times_R \mu}\right) \circ x = \lambda \circ \left({\mu \circ x}\right)$

### Right Module

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

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

A right module over $R$ is an $R$-algebraic structure $\left({G, +_G, \circ}\right)_R$ with one operation $\circ$, the (right) ring action, which satisfies the right module axioms:

 $(1)$ $:$ $\displaystyle \forall \lambda \in R: \forall x, y \in G:$ $\displaystyle \left({x +_G y}\right) \circ \lambda = \left({x \circ \lambda}\right) +_G \left({y \circ \lambda}\right)$ $(2)$ $:$ $\displaystyle \forall \lambda, \mu \in R: \forall x \in G:$ $\displaystyle x \circ \left({\lambda +_R \mu}\right) = \left({x \circ \lambda}\right) +_G \left({x\circ \mu}\right)$ $(3)$ $:$ $\displaystyle \forall \lambda, \mu \in R: \forall x \in G:$ $\displaystyle x \circ \left({\lambda \times_R \mu}\right) = \left({x \circ \lambda}\right) \circ \mu$

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

### Scalar

The elements of the scalar ring $\left({R, +_R, \times_R}\right)$ are called scalars.

### Vector

The elements of $\left({G, +_G}\right)$ are called vectors.

### Zero Vector

The identity of $\left({G, +_G}\right)$ is usually denoted $\mathbf 0$, or some variant of this, and called the zero vector.

Note that on occasion it is advantageous to denote the zero vector differently, for example by $e$, or $0_V$ or $0_G$, in order to highlight the fact that the zero vector is not the same object as the zero scalar.

## Also defined as

Sources who only deal with rings with unity often define a module as what on $\mathsf{Pr} \infty \mathsf{fWiki}$ 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

• Results about modules can be found here.