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.