# Correspondence between the ideals of a ring R and its localization at a submonoid S

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?

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.
• 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. – Bernard Jun 4 '17 at 23:40
• 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 $\mathfrak I=\varphi^{-1}(J)$. – Bernard Jun 4 '17 at 23:56