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Proposition 9.2. Let $A$ be a Noetherian local domain of dimension one, $m$ its maximal ideal,$k=A/m$ its residue field. Then the following are equivalent :

ii) $A$ is integrally closed;

iii) $m$ is a principal ideal;

Proof. Let $a\in m$ and $a\neq 0$. By remark (A) there exists an integer $n$ such that $m^n\subset (a)$, $m^{n-1}\nsubseteq (a)$. Choose $b\in m^{n-1}$ and $b\notin (a)$, and let $x=a/b\in K,$ the fraction field of $A$. We have $x^{-1}\notin A$(since $b\notin (a)$),hence $x^{-1}$ is not integral over $A$, and therefore by (5.1) we have $x^{-1}m\nsubseteq m$(for if $x^{-1}m\subset m$, m would be a faithful A[x^{-1}]-module, finitely generated as an $A-module$). But $x^{-1}m\subset A$ by construction of $x$ ,hence $x^{-1}m=A$ and therefore $m=Ax=(x)$.

The proof puzzles me is that we got $x^{-1}m \subset A$,then $x^{-1}m = A$. Why is that $x^{-1}m = A$ ? May I need to show that $m \subsetneqq x^{-1}m$ (But I think this is diffcult for me.Could you give me some hints), then the conclusion is followed by that $m$ is a maximal ideal?

Zeldovich Yakov
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  • The point is that $x^{-1}m$ is an ideal in $A$. Since $A$ is local, either $x^{-1}m=A$ or $x^{-1}m$ is contained in $m$. – Marktmeister Apr 20 '21 at 12:48
  • If $x^{-1} \in A$, then $x^{-1}m $ is an ideal of $A$. In this case $x^{-1}\notin A$, I need to show that it is an ideal. – Zeldovich Yakov Apr 20 '21 at 12:53
  • I fail to see the question. That it is an ideal follows immediately since $m$ is an ideal and since $x^{-1}m$ is contained in $A$, as you wrote. Could you be more precise, please? – Marktmeister Apr 20 '21 at 13:03
  • If $x^{-1}∈A$, then $x^{-1}m$ is an ideal of A.But in this case $x^{-1}∉A$ (actually $x^{-1}\in K-A$ ), I need to show that $x^{-1}m$ is an ideal by definition. And I think I didn't give a precise description before – Zeldovich Yakov Apr 20 '21 at 13:56
  • Do you agree that $x^{-1} m$ is an $A$-submodule of $A$? If no, try to prove it. If there is any problem, please tell me exactly where you're stuck. – Marktmeister Apr 20 '21 at 14:31
  • Thanks for your hints! – Zeldovich Yakov Apr 20 '21 at 14:47

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