Vector Space over Division Subring is Vector Space

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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