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?

  • $\begingroup$ 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 '14 at 19:30

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 '14 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 '14 at 3:19

If S is a multiplicative subset not containing zero, there is a 1-1 correspondence between primary (where $xy \in I$ such that $x \notin I$ implies $y^n \in I$ for some integer $n$ ) ideals that do not intersect S and $S^{-1}I$


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