On the residual nilpotence of free products of nilpotent groups with central amalgamated subgroups

Let $G$ be a free product of nilpotent groups $A$ and $B$ with proper amalgamated subgroups $H$ and $K$. We state that if $H$ and $K$ lie in the centers of $A$ and $B$, respectively, then $G$ is residually nilpotent if and only if the ordinary free product of $A/H$ and $B/K$ possesses the same property. We also prove that if $\pi$ is a non-empty set of primes, $H$ is central in $A$, and $K$ is normal in $B$, then $G$ is residually $\pi$-finite nilpotent if and only if $G$ is residually $\pi$-finite and the free product of $A/H$ and $B/K$ is residually $\pi$-finite nilpotent. We obtain two corollaries of the second result for the cases when $A$ and $B$ have finite ranks or finite numbers of generators. In particular, we prove that if $A$ and $B$ are finitely generated, $H$ is central in $A$, and $K$ is normal in $B$, then $G$ is residually $\pi$-finite nilpotent if and only if the periodic parts of $A$ and $B$ are $\pi$-groups and the periodic parts of $A/H$ and $B/K$ are $p$-groups for some prime $p$ which belongs to $\pi$.