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Let $f:\mathbb{C}[x,y] \to \mathbb{C}[t]$ be a homomorphism that is identity on $\mathbb{C}$ and sends $x\to x(t),y \to y(t)$ and such that $x(t),y(t)$ aren't both constant. Prove $\ker(f)$ is a principal ideal.

user26857
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dragoboy
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  • Please provide some background. Is this an exercise in a book? Homework? Is there any context? – Martin Brandenburg Sep 15 '14 at 20:54
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    Yes, Artin... No not homework, I was just trying those problems given as excercise in Artin – dragoboy Sep 15 '14 at 20:59
  • There are some other adjectives you can attach to ideals besides principal. Do you know any of them? Do any of them apply to $ker(f)$. (Thinking this way won't solve the whole problem, but may get you started.) – guy-in-seoul Sep 16 '14 at 23:32
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    Also, when I think about how I would answer this question, it uses some non-trivial commutative algebra facts. There may well be (indeed, surely is) a simpler proof that avoids them, but it would help to say what material is expected to be used as input. E.g. what chapter is this an exercise to? What were the main results of that chapter? Is this exercise starred, or indicated in some way to be difficult given what is known, or is it supposed to be routine? – guy-in-seoul Sep 16 '14 at 23:35
  • Can you please post your answer ? – dragoboy Sep 17 '14 at 08:04

1 Answers1

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The kernel is a prime ideal. The source has Krull dimension two, while the target is a subring of $\mathbb C[t]$ which is bigger than $\mathbb C$, and so has Krull dimension one. Thus the kernel has height one, i.e. it is minimal with respect to being non-zero. Thus it is principal (since $\mathbb C[x,y]$ is a UFD).

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