On the bar-radical of Jordan baric algebras
A baric algebra \(U\) over a field \(F\) is an algebra (not necessarily associative, commutative or finite-dimensional) having a weight function \(\omega\), that is, \(\omega: U\rightarrow F\) is a nonzero homomorphism of algebras. Then, \(\text{Ker}\,\omega\) is an ideal of codimension one which is denoted by \(\text{bar}(U)\). If \(B\subseteq U\), \(\text{bar}(B)=\{x\in B\mid \omega(x)=0\}\). Let \(B\) be a subalgebra of \(U\) such that \(B\neq\text{bar}(B)\); then \(B\) is called a baric subalgebra of \(U\) and \(B\) is a normal baric subalgebra if \(\text{bar}(B)\) is an ideal of \(U\). If \(I\subseteq\text{bar}(B)\) is an ideal of \(B\), then \(I\) is called a baric ideal of \(B\). A baric algebra \(U\) is \(b\)-simple if for all normal baric subalgebras \(B\) of \(U\) either \(\text{bar}(B)=0\) or \(\text{bar}(B)=\text{bar}(U)\). The bar-radical of a baric algebra \(U\), denoted by \(\text{rad}(U)\), is zero if \(U\) is \(b\)-simple, and is the intersection of \(\text{bar}(B)\) where \(B\) runs over all normal maximal baric subalgebras otherwise. The authors [Algebras Groups Geom. 21, No. 4, 387--397 (2004; Zbl 1130.17308)] have studied Jordan baric algebras previously. If \(U\) is a Jordan algebra, the nil-radical \(R(U)\) of \(U\) is defined as the maximal nil-ideal of \(U\). Then the authors proved that if \(U\) is a finite-dimensional unital Jordan baric algebra then \(\text{rad}(U)=R(U)\cap(\text{bar}(U))^2\). An example showing that the unity element is necessary is given. In the paper under review, the authors consider finite dimensional Jordan baric algebras \(U\) such that \(\text{rad}(U) \subseteq (\text{bar}(U))^3\). It is proved that if \(U\) is such an algebra, then \(\text{rad}(U)=R(U)\cap(\text{bar}(U))^3\). Moreover, conditions so that \(\text{rad}(U) \subseteq (\text{bar}(U))^3\) are given, and an example showing that such conditions are necessary, is constructed.
scientific article; zbMATH DE number 5285852