I could not get an answer to this question in MathStackExchange, so I dare ask it here.

Given any two fields, $\rm F_1,F_2$ over the same prime subfield $\rm F$, the quotient $\rm \mathbf F=F_1\otimes_F F_2/\mathcal M$ of the tensor product $\rm F_1\otimes_F F_2$ by a maximal proper ideal $\mathcal M$ provides a field with two embeddings $$\rm F_1\rightarrow\mathbf F,\ f\mapsto f\otimes 1 +\mathcal M\quad\text{and}\quad F_2\rightarrow\mathbf F,\ f\mapsto 1\otimes f +\mathcal M.$$

Question. Given two division rings $\rm R_1,R_2$ having the same characteristic, is there a way to find a division ring $\rm R$ with two embedings $\rm R_1⊂R$ and $\rm R_2⊂R$ ?