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let $k$ be a field and let $L$ be a $k$-algebra of dimension $2$. If $b \in L$ such that is not in $Vec(1_L)$, show that there exists a unique algebra morphism $k[X] \to b$ such that sending $X \to b$.

It exercise is of my course of representation, however, I don't understand good.

I think that if in fact, this morphism exists then necessarily it send $ p(X) \to p(b)$ however I am not sure to this affirmation.

could you give me a hint, please?

jfruizc273
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  • The only worry is that $b$ should commute with all the elements of $k$ and all its powers. If that is clear to you, then you can go ahead and use the (correct) idea. Mind you, there are occasions, when you need to be careful, read this excellent post by Arturo Magidin. Those concerns don't apply here because $k$ is commutative, but that is still a good read. – Jyrki Lahtonen Sep 28 '20 at 18:09

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Sending $p(X) \in k[X]$ to $p(b) \in k$ is indeed the way to go. Try to show this defines an algebra morphism. Unicity is clear as $X \to b$ is a demand.

Henno Brandsma
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