On the Residual Finiteness of Some Generalized Products of Soluble Groups of Finite Rank

Let $G$ be a free product of residually finite virtually soluble groups $A$ and $B$ of finite rank with an amalgamated subgroup $H$, $H \not= A$ and $H \not= B$. And let $H$ contains a subgroup $W$ of finite index which is normal in both $A$ and $B$. We prove that the group $G$ is residually finite if and only if the subgroup $H$ is finitely separable in $A$ and $B$. Also we prove that if all subgroups of $A$ and $B$ are finitely separable in $A$ and $B$, respectively, all finitely generated subgroups of $G$ are finitely separable in $G$.