Subring Module is Module/Special Case

Theorem
Let $S$ be a subring of the ring $\struct {R, +, \circ}$.

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

Then $\struct {R, +, \circ_S}_S$ is an $S$-module.

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

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


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


 * If $\struct {R, +, \circ}$ has a unity, then $\struct {R, +, \circ}_R$ is unitary.

Now the theorem follows directly from Subring Module.