# 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 $\displaystyle \forall \lambda \in R: \forall x, y \in G:$ $\displaystyle \lambda \circ \paren {x +_G y}$ $\displaystyle =$ $\displaystyle \paren {\lambda \circ x} +_G \paren {\lambda \circ y}$ $(\text M 2)$ $:$ Scalar Multiplication (Right) Distributes over Scalar Addition $\displaystyle \forall \lambda, \mu \in R: \forall x \in G:$ $\displaystyle \paren {\lambda +_R \mu} \circ x$ $\displaystyle =$ $\displaystyle \paren {\lambda \circ x} +_G \paren {\mu \circ x}$ $(\text M 3)$ $:$ 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:

 $(\text {RM} 1)$ $:$ Scalar Multiplication Right Distributes over Module Addition $\displaystyle \forall \lambda \in R: \forall x, y \in G:$ $\displaystyle \paren {x +_G y} \circ \lambda$ $\displaystyle =$ $\displaystyle \paren {x \circ \lambda} +_G \paren {y \circ \lambda}$ $(\text {RM} 2)$ $:$ Scalar Multiplication Left Distributes over Scalar Addition $\displaystyle \forall \lambda, \mu \in R: \forall x \in G:$ $\displaystyle x \circ \paren {\lambda +_R \mu}$ $\displaystyle =$ $\displaystyle \paren {x \circ \lambda} +_G \paren {x\circ \mu}$ $(\text {RM} 3)$ $:$ 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.