Artin's generalized theorem on alternative rings

Statement
Suppose $$R$$ is an alternative ring and $$x,y,z$$ are (possibly equal, possibly distinct) elements of $$R$$. Then the following are equivalent, where $$a$$ denotes the associator:


 * $$a(x,y,z) = 0$$
 * $$a(x,y,z) = a(x,z,y) = a(y,x,z) = a(y,z,x) = a(z,x,y) = a(z,y,x) = 0$$
 * The subring of $$R$$ generated by $$x,y,z$$ is associative.

Facts used

 * 1) uses::Subset version of Artin's generalized theorem on alternative rings

Applications

 * Artin's theorem on alternative rings