Abstract :
Many papers in connection with power associativity in genetic algebras show a class of commutative power-associative algebras which are one-dimensional modulo their maximal nil ideals. In this paper we study power-associative algebras with principal and absolutely primitive idempotent and the Peirce decomposition A = A1 circled plus A1/2 circled plus A0 of which either A1 is isomorphic to the ground field of A0 = 0. In the first case, this class of algebras, which we call power-associative image-algebras, coincide with the class of Berstein algebras of order n (n greater-or-equal, slanted 0) which are power-associative. Every power-associative image-algebra is a train algebra, and when it is a Jordan image-algebra, it is special train algebra. In the other case, we refer to power-associative algebras of type II. These algebras are also train algebras.
