I was wondering, is the following analogue of the primitive element theorem true for division rings: Let $R,S$ be division rings of characteristic $0$ such that $R$ is finite dimensional as an $S$-module. Then, there are two elements $a,b \in R$ such that $R=S(a,b)$, the sub-ring generated by elements of $S$ and the two elements $a,b$.
I was mainly trying this for $S=Q$, trying to prove or disprove that if $S=Q$, then such a ring $R$ can be embedded into the quarternions.
(Edit: Added characteristic $0$ condition to avoid the separability issue as pointed by below comment).
Edit: Found out that the second part of my question is not true. We can construct counterexamples of dimension $4$. ...