Let R be a ring such
that in every subring generated by two elements the following identities are
satisfied:
(x,y2,x)
= y ∘ (x,y,x)
(1)
(x,y,z) + (y,z,x) + (z,x,y)
= 0
(2)
((x,y),x,x)
= 0,
(3)
where (a,b,c) = (ab)c − a(bc),(a,b) = ab − ba, and a ∘ b = ab + ba. This
condition is satisfied by any alternative ring and also by any commutative
ring. Assume further that R is a simple ring of characteristic not 2 or 3
and that R has an idempotent e such that (e,e,R) = 0 = (R,e,e) while
(e,R)≠0. It is proved in this paper that under these conditions R must be
alternative.
