# Definition:Associator

## Definition

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring.

Consider the trilinear mapping $\left[{\cdot, \cdot, \cdot}\right]: A_R^3 \to A_R$ defined as:

$\forall a, b, c \in A_R: \left[{a, b, c}\right] := \left({a \oplus b}\right) \oplus c - a \oplus \left({b \oplus c}\right)$

Then $\left[{\cdot, \cdot, \cdot}\right]$ is known as the associator of $\left({A_R, \oplus}\right)$.

It can be considered a measure of how much associativity of $\oplus$ fails in $\left({A_R, \oplus}\right)$.

Note that trivially if $\left({A_R, \oplus}\right)$ is an associative algebra, then:

$\forall a, b, c \in A_R: \left[{a, b, c}\right] = \mathbf 0_R$