### 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.

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.

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