Ring is Module over Itself

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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 $\circ$.


Proof 2

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

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

where $n = 1$.