Ring is Module over Itself/Proof 1

Proof
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:

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

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