Vector Space over Division Subring is Vector Space

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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 is by definition a unitary module over a division ring.


$S$ is a division ring by assumption.

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

$\blacksquare$


Sources