ON POLYNOMIAL IDENTITIES ON TRAIN ALGEBRAS OF RANK 3.

Abstract

It is known that every train algebra of rank 2 is a Jordan algebra. This is not true for train algebras of rank 3. In this paper we prove that every train algebra of rank 3 satisfying a polynomial identity of degree three is an associative algebra. We also prove that a train algebra of rank 3 over fields of characteristic not 2,3,5 that satisfies a polynomial identity of degree four is either a Jordan algebra or satisfies the identity (xy)(xy)− (xx)(yy)=0 . Moreover, in these algebras this last identity does not oppose or imply the Jordan identity.