Division Ring is Vector Space over Prime Subfield

Theorem
Let $\left({K, +, \times}\right)$ be a division ring.

Let $\left({S, +, \times}\right)$ be the prime subfield of $K$

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