Left alternative ring

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 defining ingredient::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 $$R$$.