Division Ring is Vector Space over Prime Subfield

Theorem
Let $\struct {K, +, \times}$ be a division ring.

Let $\struct {S, +, \times}$ be the prime subfield of $K$

Then $\struct {K, +, \times_S}_S$ is an $S$-vector space, where $\times_S$ is the restriction of $\times$ to $S \times K$.