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Let $A$ be a commutative ring, and $S$ a mutiplicatively closed subset. In my text book, it is stated that:

there is one to one correspondence of prime ideals in ring $A$ (not meeting $S$) and prime ideals in its localization $S^{-1}A$.

And my question is if we can remove the word prime and state an 1-1 relation to any ideal?

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  • $\begingroup$ Which book? I would expect any book to mentioning this to have a discussion about which ideals are extended/contracted. Come to think of it, I don't remember Lang doing this... $\endgroup$
    – Hoot
    Dec 13, 2014 at 19:30

1 Answer 1

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Consider the ideals $(x)$ and $(xy)$ in the ring $k[x,y]$ and its localization $k[x,y]_{(x)}$.

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    $\begingroup$ can you give more detail about it? I'm not quite familiar with that $\endgroup$
    – annimal
    Dec 13, 2014 at 19:22
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    $\begingroup$ @annimal There are two things you should be concerned about: why $(x) \neq (xy)$ in $k[x,y]$ and why $(x) = (xy)$ in $k[x,y]_{(x)}$. Which of these would you like more clarification on? $\endgroup$
    – RghtHndSd
    Dec 14, 2014 at 3:19

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