Let $(F,v)$ be a valued field, $(K_1,v\vert_{K_1})$ and $(K_2,v\vert_{K_2})$ be two complete valued subfields of $(F,v)$. Let $R_1$ (resp. $R_2$) be a dense (w.r.t. to the topology induced by $v$) subring of $K_1$ (resp. $K_2$).
My question is: is $R_1 R_2$ dense in $K_1 K_2 \subseteq F$? By $R_1 R_2$ I mean the intersection of all subrings of $F$ containing both $R_1$ and $R_2$, i.e. the subring generated by $R_1$ and $R_2$.
If $K_1$ and $K_2$ are both finite extensions of certain common subfield, then the above assertion is true, for every element of $K_1 K_2$ can be written in the form $x=\sum_{i=1}^n a_i b_i$, where $a_i\in K_1$ and $b_i\in K_2$. However the same argument does not hold for general extensions.
