Let R be a ring
satisfying the following three defining relations: (i) (x,y2,x) = y ∘ (x,y,x),
(ii) (x,y,z) + (y,z,x) + (z,x,y) = 0, and (iii) ((x,y),x,x) = 0, where
(a,b,c) = (αb)c − a(bc), (a,b) = αb − ba, and a ∘ b = ab + ba. All three identities
follow from commutativity, hence are true in Jordan rings. Besides (i) holds in Lie
and alternative rings, (ii) holds in Lie and quasiassociative rings and in alternative
rings of characteristic three, while (iii) holds in right alternative rings. The main
result is that if R has characteristic ∓2,3 (that means no elements in R have additive
order two or three) and no divisors of zero then R must be either associative or
commutative.
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