Algebra theory MOC

Algebra homomorphism

Let $𝐴$ and $𝐵$ be $𝑅$ -algebras An algebra homomorphism $𝑓 : 𝐴 \to 𝐵$ is a $𝑅$ -linear map which preserves the bilinear product, #m/def/ralg i.e. for $𝑥, 𝑦 \in 𝐴$ and $𝜆, 𝜇 \in 𝑅$

$𝑓 (𝜆 𝑥 + 𝜇 𝑦) = 𝜆 𝑓 (𝑥) + 𝜇 𝑓 (𝑦)$
$𝑓 (𝑥 𝑦) = 𝑓 (𝑥) 𝑓 (𝑦)$

a unital algebra homomorphism is one which additionally has

$𝑓 (1 𝐴) = 1 𝐵$

Properties

Kernel of an algebra homomorphism

Special cases

Lie algebra homomorphism

#state/tidy | #lang/en | #SemBr