Subring Module is Module/Special Case

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

Let $\circ_S$ be the restriction of $\circ$ to $S \times R$.

Then $\left({R, +, \circ_S}\right)_S$ is an $S$-module.

If $\left({R, +, \circ}\right)$ has a unity,$1_R$, and $1_R \in S$, then $\left({R, +, \circ_S}\right)_S$ is a unitary $S$-module.

Proof
From Ring is Module over Itself, it follows that:

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

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

Now the theorem follows directly from Subring Module.