On a class of baric algebras Original Research Article

It is known that baric algebras satisfying the identity (x2)2 = w(x)x3 have idempotent elements and every linear form w: A → K is a multiplicative map. We prove that these algebras are Jordan-Bernstein of order 2 and special train algebras. Moreover, as a corollary we obtain that the train equation of these algebras is x4 − w(x)x3 = 0, and we give examples of baric algebras satisfying x4 − w(x)x3 = 0 but not satisfying (x2)2 = w(x)x3.