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.
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.