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Let $F$ be an algebraically closed field, and consider the ring $F[X, Y]$ of polynomials over $F$ in two indeterminates $X$ and $Y$. Let $S$ be the multiplicatively closed set in $F[X, Y]$ generated by polynomials of the form $X−α$ and $Y−β$ for all non-zero elements $α, β \in F$, and set $R=S^{-1}F[X,Y]$ a Noetherian domain of Krull dimension two. The only non-principal maximal ideal of $R$ is $M=S^{-1}(X,Y)$, while $R$ has infinitely many principal (height one) maximal ideals.

Why The only non-principal maximal ideal of $R$ is $M=S^{-1} (X,Y)$?

why $R$ has infinitely many principal (height one) maximal ideals?

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  • $\begingroup$ What makes you think these things are true? $\endgroup$
    – Steven Landsburg
    Jul 15, 2013 at 6:17
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    $\begingroup$ This is a quote from the following paper: ams.org/journals/tran/2011-363-07/S0002-9947-2011-05249-9 When quoting works, it is very important to cite the source. $\endgroup$
    – Karl Schwede
    Jul 15, 2013 at 6:38
  • $\begingroup$ This question appears to be off-topic because neither cites its sources nor provides motivation for the question. $\endgroup$
    – Theo Johnson-Freyd
    Jul 16, 2013 at 3:08

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I'm not sure if this question is appropriate for this site, but I'm going to give it the benefit of the doubt.

Look up an introductory text on commutative algebra and see what the prime ideals of $S^{-1} A$ correspond to in terms of $A$. ($A$ is a ring and $S$ is a multiplicative system).

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