# Category:Examples of Modules

This category contains examples of Module.

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 $\ds \forall \lambda, \mu \in R: \forall x \in G:$ $\ds x \circ \paren {\lambda \times_R \mu}$ $\ds =$ $\ds \paren {x \circ \lambda} \circ \mu$

## Subcategories

This category has the following 4 subcategories, out of 4 total.

## Pages in category "Examples of Modules"

The following 3 pages are in this category, out of 3 total.