On a generalization of alternative and Lie rings

Abstract:

Alternative as well as Lie rings satisfy all of the following four identities: (i) $({x^2},y,z) = x(x,y,z) + (x,y,z)x$, (ii) $(x,{y^2},z) = y(x,y,z) + (x,y,z)y$, (iii) $(x,y,{z^2}) = z(x,y,z) + (x,y,z)z$, (iv) $(x,x,x) = 0$, where the associator $(a,b,c)$ is defined by $(a,b,c) = (ab)c - a(bc)$. If $R$ is a ring of characteristic different from two and satisfies (iv) and any two of the first three identities, then it is shown that a necessary and sufficient condition for $R$ to be alternative is that whenever $a,b,c$ are contained in a subring $S$ of $R$ which can be generated by two elements and whenever ${(a,b,c)^2} = 0$, then $(a,b,c) = 0$.