On extension of classical Baer results to Poisson algebras

Abstract

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.