# Definition:Module

## Contents

## 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 both a left module and a right module:

### Left Module

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.

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

\((M \, 1)\) | $:$ | Scalar Multiplication (Left) Distributes over Module Addition | \(\displaystyle \forall \lambda \in R: \forall x, y \in G:\) | \(\displaystyle \lambda \circ \paren {x +_G y} \) | \(\displaystyle = \) | \(\displaystyle \paren {\lambda \circ x} +_G \paren {\lambda \circ y} \) | ||

\((M \, 2)\) | $:$ | Scalar Multiplication (Right) Distributes over Scalar Addition | \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\) | \(\displaystyle \paren {\lambda +_R \mu} \circ x \) | \(\displaystyle = \) | \(\displaystyle \paren {\lambda \circ x} +_G \paren {\mu \circ x} \) | ||

\((M \, 3)\) | $:$ | Associativity of Scalar Multiplication | \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\) | \(\displaystyle \paren {\lambda \times_R \mu} \circ x \) | \(\displaystyle = \) | \(\displaystyle \lambda \circ \paren {\mu \circ x} \) |

### Right Module

Let $\struct{R, +_R, \times_R}$ be a ring.

Let $\struct{G, +_G}$ be an abelian group.

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

\((RM \, 1)\) | $:$ | Scalar Multiplication Right Distributes over Module Addition | \(\displaystyle \forall \lambda \in R: \forall x, y \in G:\) | \(\displaystyle \paren {x +_G y} \circ \lambda \) | \(\displaystyle = \) | \(\displaystyle \paren {x \circ \lambda} +_G \paren {y \circ \lambda} \) | ||

\((RM \, 2)\) | $:$ | Scalar Multiplication Left Distributes over Scalar Addition | \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\) | \(\displaystyle x \circ \paren {\lambda +_R \mu} \) | \(\displaystyle = \) | \(\displaystyle \paren {x \circ \lambda} +_G \paren {x\circ \mu} \) | ||

\((RM \, 3)\) | $:$ | Associativity of Scalar Multiplication | \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\) | \(\displaystyle x \circ \paren {\lambda \times_R \mu} \) | \(\displaystyle = \) | \(\displaystyle \paren {x \circ \lambda} \circ \mu \) |

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

### Scalar

The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called **scalars**.

### Vector

The elements of $\struct {G, +_G}$ are called **vectors**.

### Zero Vector

The identity of $\struct {G, +_G}$ 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.

## Left vs 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:

But this is not the case for a ring that is not commutative. From:

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

From:

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

From:

a left module induces a right module and vice-versa if and only if actions are commutative.

## 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.

### Special cases

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 26$