\begin{definition}[Definition:Associator]
Let $\struct {A_R, \oplus}$ be an algebra over a ring.
Consider the trilinear mapping $\sqbrk {\cdot, \cdot, \cdot}: A_R^3 \to A_R$ defined as:
:$\forall a, b, c \in A_R: \sqbrk {a, b, c} := \paren {a \oplus b} \oplus c - a \oplus \paren {b \oplus c}$
Then $\sqbrk {\cdot, \cdot, \cdot}$ is known as the '''associator''' of $\struct {A_R, \oplus}$.
It can be considered a measure of how much associativity of $\oplus$ fails in $\struct {A_R, \oplus}$.
{{ExtractTheorem|below}}
Note that trivially if $\struct {A_R, \oplus}$ is an associative algebra, then:
:$\forall a, b, c \in A_R: \sqbrk {a, b, c} = \mathbf 0_R$
\end{definition}