I hope that the following questions can be solved by Ring Theory concepts as I'm not studying further yet.

Let $A$ be an integral domain and $K$ its field of fractions. An integral domain $A$ is said to be integrally closed if $A$ satisfies the following condition: for $\alpha \in K$ if there exists a monic polynomial $f(X)$ whose coefficients are in $A$ satisfying $f(\alpha)=0$, then $\alpha \in A$.

My first question is to prove that if an integral domain $A$ is integrally closed, then the ring of fractions $S^{-1}A$ of $A$ with respect to a multiplicatively closed subset $S$ of $A$ is also integrally closed.

My attempt: Since $A$ is integrally closed, we have the integral closure of $A$ is $A$ itself. It is a theorem that (see here for instance) $S^{-1}A$ is the integral closure of $S^{-1}A$ in $S^{-1}K=K$. It follows that $S^{-1}A$ is also integrally closed.

Is this a correct proof?

My second question is trying to check whether each of the following integral domains $A$ is integrally closed or not.

(a) $A=\mathbb C[X,Y]/(X^{n}-Y^{m})$, where $n>m$ be coprime positive integers.

(b) $A=\mathbb C[X,Y]/(XY-1)$.

I'm might be aware of events that both of these above integral domains are not UFDs but I have no idea about their integrally closedness. Any help would be much appreciated

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    $\begingroup$ The answer to the first question is positive. (But please don't do this again and try to post one question at a time.) $\endgroup$ – user26857 Oct 5 '15 at 13:51
  • $\begingroup$ @user26857 Thanks for the answer. I posted the double since I thought that the first question may be applied to solve the second one. $\endgroup$ – user Oct 5 '15 at 14:08
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    $\begingroup$ Well, this could be a reason. And yes, one can use it for proving (b). $\endgroup$ – user26857 Oct 5 '15 at 14:40
  • $\begingroup$ @user26857 That would be great if you provide the proof for (b) by using the first question. $\endgroup$ – user Oct 5 '15 at 14:47
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    $\begingroup$ Isn't this obvious? $\mathbb C[X]$ is integrally closed (it's a UFD), and then so are its rings of fractions. $\endgroup$ – user26857 Oct 5 '15 at 15:14

(a) $\mathbb C[X,Y]/(X^{n}-Y^{m})\simeq\mathbb C[T^m,T^n]$, where $n>m$ are coprime positive integers. The field of fractions of $\mathbb C[T^m,T^n]$ is $\mathbb C(T)$ (why?), $T$ is integral over $\mathbb C[T^m,T^n]$, and $T\notin\mathbb C[T^m,T^n]$.

(b) $\mathbb C[X,Y]/(XY-1)\simeq S^{-1}\mathbb C[X]$, where $S=\{1,X,X^2,\dots\}$.

  • $\begingroup$ I may ask stupid questions but there were holes in my study. For (a), why $\mathbb C[X,Y]/(X^{n}-Y^{m})\simeq\mathbb C[T^m,T^n]$? I cannot answer your question that why the field of fractions of $\mathbb C[T^m,T^n]$ is $\mathbb C(T)$? Finally, why $\mathbb C[X,Y]/(XY-1)\simeq S^{-1}\mathbb C[X]$? $\endgroup$ – user Oct 5 '15 at 15:34
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    $\begingroup$ Well, it seems the comments could be much longer than the answer. (a) Define a map from $\mathbb C[X,Y]$ to $\mathbb C[T]$ by sending $X$ to $T^m$ and $Y$ to $T^n$. Then look at its image an kernel. (On the way you may want to show that $X^{n}-Y^{m}$ is irreducible. (b) $\mathbb C[X,Y]/(XY-1)=\mathbb C[X,X^{-1}]$. $\endgroup$ – user26857 Oct 5 '15 at 15:56
  • $\begingroup$ Yes, you're right. Your map can prove that $\mathbb C[X,Y]/(X^{n}-Y^{m})$ isomorphic to a subring of the integral domain $\mathbb C[T]$. So the field of fractions of $\mathbb C[X,Y]/(X^{n}-Y^{m})$ is $\mathbb C(T)$. But does it prove that $\mathbb C[X,Y]/(X^{n}-Y^{m})\simeq\mathbb C[T^m,T^n]$? $\endgroup$ – user Oct 5 '15 at 16:10
  • $\begingroup$ Thanks for giving me the right track. Quick search gives me the desired answers. $\endgroup$ – user Oct 5 '15 at 16:23

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