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Suppose $A$ be a commutative ring with unity. $S$ be a multiplicative closed subset of $A$. Suppose it is given that $A_S$ is a local ring. What can we say about $S$. Is it true that $S$ must be a complement of a prime ideal?

Thank you.

  • What does $A_S$ denote? – M. Winter Dec 14 '17 at 09:07
  • I think It means localization of $S$. – 1ENİGMA1 Dec 14 '17 at 09:52
  • You can find your answer in there: https://math.stackexchange.com/questions/1088469/is-the-complement-of-a-prime-ideal-closed-under-both-addition-and-multiplication – 1ENİGMA1 Dec 14 '17 at 09:52
  • It seems dublicated. – 1ENİGMA1 Dec 14 '17 at 09:53
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    @1ENİGMA1 That question assumes $S$ is the complement of a prime ideal, and asks if $S$ is multiplicatively closed. This question assumes $S$ is multiplicatively closed and that $R_S$ is local, and asks about $R\setminus S$. So... it is clearly not a duplicate of that question. But I have a feeling it still might be a duplicate of a different question. – rschwieb Dec 14 '17 at 14:35
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    I found the duplicate. – rschwieb Dec 14 '17 at 14:38

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