Associator on a non-associative ring

Definition
Suppose $$R$$ is a non-associative ring (i.e., a not necessarily associative ring). The associator on $$R$$ is defined as the function:

$$a: R \times R \times R \to R$$

given by:

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

Here, $$-$$ is the subtraction operation corresponding to the additive group of $$R$$ and $$*$$ is the multiplication on $$R$$.

Facts

 * $$a$$ is the zero function if and only if $$R$$ is a non-associative ring.
 * $$a$$ is an alternating function in its variables if and only if $$R$$ is an alternative ring.
 * $$a$$ is alternating in the first two variables if and only if $$R$$ is a left-alternative ring.
 * $$a$$ is alternating in the last two variables if and only if $$R$$ is a right-alternative ring.
 * $$a$$ is alternating in the first and last variable if and only if $$R$$ is a flexible ring.
 * $$a$$ is additive in each variable. Further, if $$R$$ is an algebra over a field $$k$$, then $$a$$ is $$k$$-linear in each variable.
 * The left kernel of $$a$$ is the set of elements $$x \in R$$ such that $$a(x,y,z) = 0$$ for all $$y,z \in R$$. This coincides precisely with the set of left-associative elements of $$R$$, and is a subring of $$R$$ called the left nucleus.
 * The middle kernel of $$a$$ is the set of elements $$y \in R$$ such that $$a(x,y,z) = 0 $$ for all $$x,z \in R$$. This coincides precisely with the set of middle-associative elements of $$R$$, and is a subring of $$R$$ called the middle nucleus.
 * The right kernel of $$a$$ is the set of elements $$z \in R$$ such that $$a(x,y,z) = 0$$ for all $$x,y \in R$$. This coincides precisely with the set of right-associative elements of $$R$$ and is a subring of $$R$$ called the right nucleus.

The associator also satisfies an identity called the associator identity with four (universally quantified) variables and five terms, which is closely related to the associativity pentagon:

$$\! w * a(x,y,z) + a(w,x,y) * z = a(w*x,y,z) - a(w,x*y,z) + a(w,x,y*z)$$

Related notions

 * The additive commutator of a (possibly non-associative ring), defines as $$(x,y) \mapsto (x * y) - (y * x)$$, plays a similar role for commutativity.
 * The Kleinfeld function builds upon the associator and is used to prove facts about alternative rings.