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.
Special Case
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.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.3$