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I am studying the localization of a ring R at a submonoid S. I am really confused of the form of the ideals in the localization.

In the case of the quotient ring A by an ideal I, it is defined a bijection between the ideals of A containg I and the ideals of A/I.

Is this rule true anymore in the localization of a ring at a submonid?

Can we say that there exists a bijection between the ideals of the ring R that does not meet S and the ideals of the localization of the ring R at the submonid S? Can we say that every ideal of the localization of the ring R at a submonoid S is of the form Is, where I is an ideal of R that does not meet S?

May you help me, please? Thank you in advance.

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There is such a bijection between the prime ideals of $S^{-1}R$ and the prime ideals of $R$ which do not meet $S$. For not necessarily prime ideals, every ideal in $S^{-1}R$ has the form $S^{-1}\mathfrak a$ for some ideal $\mathfrak a\subset R$, but the correspondence is not necessarily injective.

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  • $\begingroup$ Thank you very much for your answer! So, you mean that if J is an ideal in S^{-1}R, then J has the form S^{-1}\mathfrak I, for some ideal I of R, but not for all the ideals I of R. Is this correspondence always surjective? I think yes, since the correspondence between R and S^{-1}R such that a goes to a/s is surjective, but I do not know if I am right. $\endgroup$ – User1999 Jun 4 '17 at 23:26
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    $\begingroup$ I mean that there may be several ideals in $R$ extending to the same ideal in $S^{-1}R$. So yes it is surjective. A reference (in the more genral context of submodules) is Bourbaki, Commutative Algebra, Ch. II, Rings and Modules of Fractions, §2, n°4, prop. 10. $\endgroup$ – Bernard Jun 4 '17 at 23:40
  • $\begingroup$ Ok, I will look for them when I'll pass to modules. Thank you! $\endgroup$ – User1999 Jun 4 '17 at 23:43
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    $\begingroup$ F.y.i., if $\varphi:R\longrightarrow S^{-1}R$ is the canonical map, the ideal in $R$ which extends to $J$ in $S^{-1}R$ is simply $\mathfrak I=\varphi^{-1}(J)$. $\endgroup$ – Bernard Jun 4 '17 at 23:56

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