> y]> z> «) = 0 for all x, y, z in A. In this paper the result of Block  and Shestakov  that a simple finite dimensional such algebra over a field of characteristic Φ 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 rings which satisfy (w, x2, z) = x-(w,x,z) and ([x,y]9yfy) = 0, and in the second case, flexible rings satisfying (w, x2
9 z) =X'(w, x, z) + (a?, x, [w, z\). Under certain conditions the rings are shown to be noncommutative Jordan or alternative espectively.