# 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:

### 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:

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

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

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

### 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:

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

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

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

**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 the abelian group $\struct {G, +_G}$ are called **vectors**.

### Zero Vector

The identity of $\struct {G, +_G}$ is usually denoted $\bszero$, or some variant of this, and called the **zero vector**:

- $\forall \mathbf a \in \struct {G, +_G, \circ}_R: \bszero +_G \mathbf a = \mathbf a = \mathbf a +_G \bszero$

Note that on occasion it is advantageous to denote the **zero vector** differently, for example by $e$, or $\bszero_V$ or $\bszero_G$, 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.

Given the remarks above, a theorem about **modules**, where the type of **module** 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 may not be true for a mix of left and right modules, unless the **modules** are over a commutative ring.

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 mix 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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

- Results about
**modules**can be found**here**.

### Special cases

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**module** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**module** - 2003: P.M. Cohn:
*Basic Algebra: Groups, Rings and Fields*: Chapter $4$: Rings and Modules : $\S 4.1$ The Definitions Recalled - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**module**