Definition:Vector Space over Division Subring/Special Case

Theorem
Let $$S$$ be a division subring of the ring with unity $$\left({R, +, \circ}\right)$$ whose unity is $$1_R$$ such that $$1_R \in S$$.

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