# Ring is Module over Itself

## Theorem

Let $\struct {R, +, \circ}$ be a ring.

Then $\struct {R, +, \circ}_R$ is an $R$-module.

### Ring with Unity

Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.

Then $\struct {R, +, \circ}_R$ is a unitary $R$-module.

## Proof 1

Note that:

$\struct {R, +, \circ}$ is a ring by assumption.

$\struct {R, +}$ is an abelian group by the definition of a ring.

Let us verify the module axioms:

 $(1)$ $:$ $\ds \forall x, y, z \in R:$ $\ds x \circ \paren {y + z} = \paren {x \circ y} + \paren {x \circ z}$ $(2)$ $:$ $\ds \forall x, y, z \in R:$ $\ds \paren {x + y} \circ z = \paren {x \circ z} + \paren {y \circ z}$ $(3)$ $:$ $\ds \forall x, y, z \in R:$ $\ds \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$

Axiom $(1)$ and $(2)$ follow from distributivity of $\circ$.

Axiom $(3)$ follows from associativity of $\circ$.

$\blacksquare$

## Proof 2

This is a special case of Module on Cartesian Product is Module:

$\struct {R^n, +, \circ}_R$ is an $R$-module

where $n = 1$.

$\blacksquare$