Definition:Commutator/Algebra

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

Consider the bilinear mapping $\left[{\cdot, \cdot}\right]: A_R^2 \to A_R$ defined as:
 * $\forall a, b \in A_R: \left[{a, b}\right] := a \oplus b - b \oplus a$

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

Note that trivially if $\left({A_R, \oplus}\right)$ is a commutative algebra, then:
 * $\forall a, b \in A_R: \left[{a, b}\right] = \mathbf 0_R$