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Let $k$ be a field of characteristic zero, $x,y$ commuting variables over $k$.

Question 1: Is it possible to characterize all ideals of $k[y,y^{-1}]$ and of $k[x,y,y^{-1}]$?

I guess that the answer to $k[y,y^{-1}]$ is simpler than the answer to $k[x,y,y^{-1}]$?

Question 2: Is it possible to characterize all maximal ideals of $k[y,y^{-1}]$ and of $k[x,y,y^{-1}]$?

Example: (1) $I= \langle \frac{x}{y},x \rangle= \langle x \rangle$ is maximal in $k[x,y,y^{-1}]$?

(2) $J= \langle x,y \rangle$ is not maximal in $k[x,y,y^{-1}]$, since $1=y^{-1}y \in \langle x,y \rangle$, so $J=k[x,y,y^{-1}]$.

Thank you very much!

user237522
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    What do you know about localisations? – LetGBeTheGraph Sep 21 '20 at 15:09
  • @LetGBeTheGraph, oh, you mean that I should consider $k[y,y^{-1}]$ as localization of $k[y]$ at $S=\{y^i\}$? and $k[x,y,y^{-1}]$ as localization of $k[x,y]$ at $S$? – user237522 Sep 21 '20 at 15:11
  • Every (maximal) ideal is the image of an (maximal) ideal of the 'original' ring? – user237522 Sep 21 '20 at 15:12
  • I would refer you to a textbook about this, it is almost a theory in itself, in particular about extensions and contractions of ideals in localisations (I can only imagine that Eisenbud treats this, everything is in that book). – LetGBeTheGraph Sep 21 '20 at 17:02
  • Thank you for your helpful hint and comment. – user237522 Sep 21 '20 at 18:39
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    The prime ideals of a localization $S^{-1}R$ are exactly the prime ideals of $R$ which do not intersect $S$. Combining this with the fact that the prime ideals of $k[x]$ are the ideals generated by prime elements of $k[x]$ since it's a PID and the ideals of $k[x,y]$ are as described in the link, you get a complete characterization. Alternatively, you may use the nullstellensatz (especially for the maximal ideals). It's kind of stunning that you are unaware of these facts after spending 5 years and ~200 posts in this tag. – KReiser Sep 21 '20 at 18:54
  • @KReiser, thanks. I wanted to make sure that what I thought was correct (more precisely, in the last two years or so I was working in areas other than math, and only here and there continued to think about math problems). – user237522 Sep 21 '20 at 19:00

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