\begin{definition}[Definition:Commutator/Algebra]
Let $\struct {A_R, \oplus}$ be an algebra over a ring.
Consider the bilinear mapping $\sqbrk {\, \cdot, \cdot \,}: A_R^2 \to A_R$ defined as:
:$\forall a, b \in A_R: \sqbrk {a, b} := a \oplus b - b \oplus a$
Then $\sqbrk {\, \cdot, \cdot \,}$ is known as the '''commutator''' of $\struct {A_R, \oplus}$.
Note that trivially if $\struct {A_R, \oplus}$ is a commutative algebra, then:
:$\forall a, b \in A_R: \sqbrk {a, b} = \mathbf 0_R$
\end{definition}