Division_Ring_is_Vector_Space_over_Prime_Subfield_v2

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 $S \times K$.