Vector Space over Division Subring is Vector Space

Theorem
Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$.

Let $S$ be a division subring of $R$, such that $1_R \in S$.

The vector space $\struct {R, +, \circ_S}_S$ over $\circ_S$ is a $S$-vector space.

Proof
A vector space over a division ring $D$ is by definition a unitary module over $D$.

$S$ is a division ring by assumption.

$\struct {R, +, \circ_S}_S$ is a unitary module by Subring Module is Module/Special Case.