Definition:Commutator/Algebra
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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:
- $\forall a, b \in A_R: \sqbrk {a, b} = \mathbf 0_R$