Equivalence of definitions of alternative ring

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 (all definitions of alternative ring) are equivalent for a non-associative ring $R$ with multiplication denoted by $*$:

1. $(R,*)$ is an 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 Left-alternative magma (?) and a 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 Right-alternative magma (?) and a 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.

Facts used

1. Alternating function condition is transitive

Proof

The associator function:

$a(x,y,z) = ((x * y) * z) - (x * (y * z))$

is additive separately in each variable, i.e., it is multi-additive. Here is what each condition says:

Condition name Equational form What does it tell us about the associator function?
left-alternative $(x * x) * y = x * (x * y)$ alternating in first and second variable
right-alternative $(x * y) * y = x * (y * y)$ alternating in second and third variable
flexible $x * (y * x) = (x * y) * x$ alternating in first and third variable

Fact (1) now tells us that any two of these conditions must imply the third, completing the proof.