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Let $\varphi: \mathbb{C}[x, y] \rightarrow \mathbb{C}[t]$ be a homomorphism that is the identity on $\mathbb{C}$ and sends $x \rightsquigarrow x(t)$ $y \rightsquigarrow y(t),$ and such that $x(t)$ and $y(t)$ are not both constant. Prove that the kernel of $\varphi$ is a principal ideal.

This comes from Artin Second Edition, page 357. The only way I know to prove someting is a principal idea is division with reminder. But this doesn't work since there're two variables.

Many thanks!

niels wen
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  • The other way is to try to guess (not randomly of course) what a generator might look like, and then prove that indeed all elements in the ideal are multiple of that generator. Try to ask yourself "imagine this is indeed a principal ideal; what would a generator need to look like?". – Captain Lama Feb 28 '21 at 12:03
  • @Ennar thanks! I didn't find this at first. – niels wen Feb 28 '21 at 12:17
  • @niels wen, you are welcome. Searching questions on MSE can be tricky sometimes. I'd recommend using https://approach0.xyz/. – Ennar Feb 28 '21 at 12:19

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