Definition:Module

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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