A note on generalizing alternative rings

Abstract

Let R R be a nonassociative ring of characteristic different from 2 2 and 3 3 which satisfies the following identities: \[ ( i) ( a b , c , d ) + ( a , b , [ c , d ] ) = a ( b , c , d ) + ( a , c , d ) b , ({\text {i)}}\;(ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b, \] \[ ( ii) ( a , a , a ) = 0 , ({\text {ii)}}\;(a,a,a) = 0, \] \[ ( iii) ( a , b ∘ c , d ) = b ∘ ( a , c , d ) + c ∘ ( a , b , d ) ({\text {iii)}}\;(a,b \circ c,d) = b \circ (a,c,d) + c \circ (a,b,d) \] for all a , b , c , d ∈ R a,b,c,d \in R and with x ∘ y = ( x y + y x ) / 2 x \circ y = (xy + yx)/2 . We prove that if R R is semiprime, then R R is alternative.