# Definition:Commutator/Algebra

## Definition

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:

$\forall a, b \in A_R: \sqbrk {a, b} = \mathbf 0_R$