Algebra homomorphism
Properties
Special cases

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, falg 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

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Algebra theory MOC
Category of algebras
Category of unital associative algebras
Clifford algebra
Exterior algebra
K-monoid
Kernel of an algebra homomorphism
Lie algebra homomorphism
Module over a unital associative algebra
Symmetric algebra
Universal enveloping algebra

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