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 $$S \times K$$.