Vol. 73, No. 1, 1977

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On generalizations of alternative algebras

Joyce Longman

Vol. 73 (1977), No. 1, 131–141

Let A be a noncommutative Jordan algebra in which ([x,y],z,z) = 0 for all x, y, z in A. In this paper the result of Block [4] and Shestakov [13] that a simple finite dimensional such algebra over a field of characteristic 2 is either alternative or Jordan is extended to the infinite dimensional case with idempotent. In the case of a noncommutative Jordan algebra satisfying the weaker identity ([x,y],y,y) = 0 for all x, y in the algebra, a simple finite dimensional such algebra is shown to be commutative, alternative, or an algebra of degree two.

In §2 we consider in the first case, power associative rlngs which satisfy (w,x2,z) = x (w,x,z) and ([x,y],y,y) = 0, and in the second case, flexible rings satisfying (w,x2,z) = x (w,x,z) + (x,x,[w,z]). Under certain conditions the rings are shown to be noncommutative Jordan or alternative respectively.

Mathematical Subject Classification 2000
Primary: 17A15
Received: 1 April 1976
Revised: 26 April 1977
Published: 1 November 1977
Joyce Longman