Kernel of an algebra homomorphism

The kernel $k e r 𝑓$ of an algebra homomorphism $𝑓 : 𝐴 \to 𝐵$ over a field $𝕂$ is simply its linear kernel, falg i.e. $k e r 𝑓 = 𝑓 - 1 {0}$ . The kernel is necessarily a (two-sided) algebra ideal of $𝐴$ .

Proof

Let $𝑘 \in 𝐾 = k e r 𝑓$ , so that $𝑓 (𝑘) = 0$ . Then for any $𝑎 \in 𝐴$ ,
$𝑓 (𝑎 \cdot 𝑘) = 𝑓 (𝑎) \cdot 𝑓 (𝑘) = 𝑓 (𝑎) \cdot 0 = 0 𝑓 (𝑘 \cdot 𝑎) = 𝑓 (𝑘) \cdot 𝑓 (𝑎) = 0 \cdot 𝑓 (𝑎) = 0$
so $𝐾 ⊴ 𝐴$ is a two-sided algebra ideal of $𝐴$ .

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Kernel of an algebra homomorphism

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