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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.
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