Left 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

Symbol-free definition

A non-associative ring (i.e., a not necessarily associative ring) is termed a left-alternative ring if it satisfies the following equivalent conditions:

1. The associator is an alternating function of its first two variables.
2. Its multiplicative magma is a left-alternative magma.

Definition with symbols

A non-associative ring $R$ (i.e., a not necessarily associative ring $R$) is termed a left-alternative ring if it satisfies the following identity:

$\!(x*x)*y=x*(x*y)\ \forall \ x,y\in R$

Note that $x,y$ are allowed to be equal. Here, $*$ is the multiplication of $R$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (Reverse implication failure)	Intermediate notions
Associative ring	associativity holds universally			|FULL LIST, MORE INFO
Alternative ring	both left-alternative and right-alternative			|FULL LIST, MORE INFO