On extension of classical Baer results to Poisson algebras

In this paper we prove that if $P$ is a Poisson algebra and the $n$th hypercenter (center) of $P$ has a finite codimension, then $P$ includes a finite-dimensional ideal $K$ such that $P/K$ is nilpotent (abelian). As a corollary, we show that if the $n$th hypercenter of a Poisson algebra $P$ (over some specific field) has a finite codimension and $P$ does not contain zero divisors, then $P$ is an abelian algebra.