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Let $R_1$ and $R_2$ are two rings such that $R_1$ $\subseteq$ $R_2$ and $R_2$ is a finitely generated ring over $R_1$. We know that, if $R_1$ is Noetherian and $S$ is a ring such that $R_1$ $\subseteq$ $S$ $\subseteq$ $R_2$ and $R_2$ is a finitely generated module over $S$, then $S$ is a finitely generated ring over $R_1$.

I found out in the June edition of The American Mathematical monthly that if we remove the condition that "$R_2$ is a finitely generated module over $S$", then there exists examples where $S$ might not be a finitely generated ring over $R_1$ and they construct an example using heavy machineries, I was thinking whether a 'simpler' example could be constructed.

And Also if one could provide a general algorithm on constructing such rings, it will be very much appreciable.

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marked as duplicate by user26857 abstract-algebra Jun 21 '17 at 6:38

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    $\begingroup$ I have no idea what it would mean to give a "general algorithm" here. $\endgroup$ – Eric Wofsey Jun 21 '17 at 4:27
  • $\begingroup$ @EricWofsey it means that my question has 2 parts, first part asks to give just an example of the fact and the second one asks to give a $general$ mechanism to construct such examples. $\endgroup$ – reflexive Jun 21 '17 at 4:30
  • $\begingroup$ The first paragraph refers to the Artin-Tate lemma. $\endgroup$ – user49640 Jun 21 '17 at 4:39
  • $\begingroup$ I am sure examples of this have been given in this site a few times — I myself have written a couple of answers, in fact! Have you looked before asking? $\endgroup$ – Mariano Suárez-Álvarez Jun 21 '17 at 5:24
  • $\begingroup$ Tip: wrapping a text with asterisks produces italics. Using double asterisks gives bold. The spacing in such italics is better than what you get with $math$ $italics$, where letters are treated as symbols (as they should). If you prefer the latter, then I apologize. We can roll my edit back. $\endgroup$ – Jyrki Lahtonen Jun 21 '17 at 6:00
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Let $R_1 = k, R_2 = k[x,y], S = k[x,xy,xy^2,xy^3,\dots]$.

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