# 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 $1_R \in S$.

## 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 is Module/Special Case.

$\blacksquare$