Alternative ring

This article defines a non-associative ring property: a property that an be evaluated to true or false for any non-associative ring.
View other non-associative ring properties

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 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.
5. The subring of $R$ generated by any two elements of $R$ is an associative ring.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions