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 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:

\((\text M 1)\)	$:$		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 $\struct {G, +_G}$ 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:

• Left Module over Commutative Ring induces Right Module
• Right Module over Commutative Ring induces Left Module.

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

• Left Module Does Not Necessarily Induce Right Module over Ring
• Right Module Does Not Necessarily Induce Left Module over Ring

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

From:

• Left Module over Ring Induces Right Module over Opposite Ring
• Right Module over Ring Induces Left Module over Opposite Ring

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

From:

• Left Module induces Right Module over same Ring iff Actions are Commutative
• Right Module induces Left Module over same Ring iff Actions are Commutative

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.

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

• Results about modules can be found here.

Special cases

• Definition:Unitary Module

• Definition:Vector Space

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
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): module