A generalization of commutative and alternative rings. IV

Abstract:

We have shown previously for rings $R$ of characteristic $\ne 2,3$ which satisfy the three identities (i) $(x,{y^2},x) = y \circ (x,y,x)$, (ii) $(x,y,z) + (y,z,x) + (z,x,y) = 0$, and (iii) ($((x,y),x,x) = 0$, where $(a,b,c) = (ab)c - a(bc),(a,b) = ab - ba$, and $a \circ b = ab + ba$, that under the assumption of no divisors of zero, all such $R$ must be either associative or commutative. Here we weaken the Lie-admissible identity (ii) by assuming instead (iv) Lie-admissibility for every subring generated by two elements. It turns out that rings without divisors of zero and of characteristic $\ne 2,3$ which satisfy (i), (iii) and (iv) are either commutative or alternative. If $S$ is a ring in which every subring generated by two elements is either commutative or associative, then identities (i), (iii) and (iv) hold in $S$, so that this result applies to $S$.