Subring Module is Module

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

Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module.

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

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

The module $\left({G, +_G, \circ_S}\right)_S$ is called the $S$-module obtained from $\left({G, +_G, \circ}\right)_R$ by restricting scalar multiplication. (Eugh, that's a bit of a mouthful. We gotta come up with something better than that.)

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

Special Case
$\left({R, +, \circ}\right)_S$ is an $S$-module where $\circ$ is the restriction of $\circ$ to $S \times R$.

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