Vector Space over Division Subring is Vector Space

Theorem
Let $\struct {L, +_L, \times_L}$ be a division ring.

Let $K$ be a division subring of $\struct {L, +_L, \times_L}$.

Let $\struct {G, +_G, \circ}_L$ be a $L$-vector space.

Let $\circ_K$ be the restriction of $\circ$ to $K \times G$.

Hence let $\struct {G, +_G, \circ_K}_K$ be the vector space induced by $K$.

Then $\struct {G, +_G, \circ_k}_k$ is indeed a $K$-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.