Ring is Module over Itself/Proof 1

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

Then $\left({R, +, \circ}\right)_R$ is an $R$-module.

If $\left({R, +, \circ}\right)$ has a unity, then $\left({R, +, \circ}\right)_R$ is unitary.

Proof
Note that:

$\left({R, +, \circ}\right)$ is a ring by assumption.

$\left({R, +}\right)$ is an abelian group by the definition of a ring.

Let us verify the module axioms:

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

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

Assume now that $\left({R, +, \circ}\right)$ has a unity, $1_R$.

For $\left({R, +, \circ}\right)_R$ to be unitary, it must satisfy the additional axiom:

The axiom follows from the definition of a unity.