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.
• ,math>a</math> 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 $R$.
• 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 $R$.
• 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 $R$.

The associator also satisfies an identity with four variables and five terms, which is closely related to the associativity pentagon. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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