Artin's theorem on alternative rings

This article gives a proof/explanation of the equivalence of multiple definitions for the term alternative ring
View a complete list of pages giving proofs of equivalence of definitions

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 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 Diassociative magma (?) under the multiplication operation.

Facts used

1. 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 $a(x,y,y)=0$.