The proof of the weak Nullstellensatz in Qing Liu’s book

Question

I am reading Qing Liu’s book and the whole proof on pp 30 goes like this:

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I do not understand the conclusion in the yellow-highlighted line. Let $f$ be the isomorphism highlighted in red. Why $f$ maps $\alpha_i$ to $\alpha_i$?

Answer 1

These corollaries (explicitly) talk about $k$-algebras, so every morphism is $k$-linear. This implies that $f(\alpha_i)=\alpha_i+\mathfrak{m}$.

In the category of $k$-algebras $k$ is an initial object, so for any $k$-algebra $A$ there is exactly one $k$-algebra homomorphism $g\colon k\to A$. This must be a ring morphism, so $g(1)=1_A$. And it must be $k$-linear, so $g(\alpha)=\alpha \cdot 1_A$.