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.