On the Bar-radical of Jordan Baric Algebras

Abstract

In this paper, we prove that if (U, w) is a finite dimensional Jordan baric algebra such that \(\text{rad}(U)\subseteq(\text{bar}(U))^3\) then, \(\text{rad}(U)=R(U)\cap(\text{bar}(U))^3\), where R(U) is the nilradical (maximal nil ideal) of U. We also give conditions so that \(\text{rad}(U)\subseteq (\text{bar}(U))^3\) and an example showing that such conditions are necessary.