The bar-radical of baric algebras (Q1924954)

The bar-radical for any baric algebra and for simple and semisimple baric algebras is defined. It is proved that the bar-radical of the join of finitely many baric algebras having an idempotent element of weight 1 is the sum of the bar-radicals of these summand algebras. It is also proved that if an Artinian baric algebra has an idempotent element of weight 1 and its bar-radical is zero, then the algebra is a semisimple baric algebra. Some examples are given.