Alternative ring

Definition
An alternative ring is a non-associative ring $$R$$ (i.e., a not necessarily associative ring) that, under its multiplication $$*$$, satisfies one of the following equivalent conditions:


 * 1) $$(R,*)$$ is an defining ingredient::alternative magma, i.e., it satisfies the identities $$x * (x * y) = (x * x) * y$$ and $$x * (y * y) = (x * y) * y$$ for all $$x,y \in R$$.
 * 2) $$(R,*)$$ is both a defining ingredient::left-alternative magma and a defining ingredient::flexible magma, i.e., it satisfies the identities $$x * (x * y) = (x * x) * y$$ and $$x * (y * x) = (x * y) * x$$ for all $$x,y \in R$$.
 * 3) $$(R,*)$$ is both a defining ingredient::right-alternative magma and a defining ingredient::flexible magma, i.e., it satisfies the identities $$x * (y * y) = (x * y) * y$$ and $$x * (y * x) = (x * y) * x$$ for all $$x,y \in R$$.
 * 4) The associator function on $$R$$ is an alternating function on any two of its variables.
 * 5) The subring of $$R$$ generated by any two elements of $$R$$ is an associative ring.

Equivalence of definitions

 * The equivalence of definitions (1) -- (4) is proved by showing that they are all equivalent to (4).
 * The equivalence with definition (5) is Artin's theorem on alternative rings.