Artin's theorem on alternative rings

Statement
The following are equivalent for a non-associative ring $$R$$ (i.e., a not necessarily associative ring $$R$$):


 * $$R$$ is an alternative ring, i.e., it is an fact about::alternative magma under its multiplication operation.
 * $$R$$ is a diassociative ring, i.e., the subring generated by any two elements is associative. Equivalently, $$R$$ is a fact about::diassociative magma under the multiplication operation.

Facts used

 * 1) uses::Artin's generalized theorem on alternative rings: This says that if $$x,y,z$$ are (possibly equal, possibly distinct) elements of an alternative ring such that $$a(x,y,z) = 0$$ where $$a$$ denotes the associator, then the subring generated by $$x,y,z$$ is associative.

Proof
Given: An alternative ring $$R$$. Elements $$x,y \in R$$ (possibly equal, possibly distinct)

To prove: The subring of $$R$$ generated by $$x$$ and $$y$$ is associative

Proof: Set $$z = y$$ and use Fact (1), along with the observation that, because $$R$$ is alternative, we must have $$a(x,y,y) = 0$$.