This is taken from Neukirch's algebraic number theory Proposition 12.3.
Proposition (12.3). If $a\neq 0$ is an ideal of an order (one dimensional Noetherian integral domain) $o$, then:
$o/a = \oplus_{p}o_p/ao_p = \oplus_{a\subseteq p}o_p/ao_p$
My question is regarding the first line of the proof: Let $\tilde{a}_p = o\cap ao_p$.
As I understand it, $o_p$ is the localization of $o$ at the prime $p$. Therefore, there are two cases:
- $a \not\subset p$: $\tilde{a}_p = o_p$. As is stated firmly in the book.
- $a \subseteq p$: $\tilde{a}_p = a$.
However, regarding case #2, the author implicitly says that $a\subseteq \tilde{a}_p \subseteq p$, and moreover, that whereas $a$ may be contained in several prime ideals $p$, $\tilde{a}_p$ is contained only in $p$. If I am not completely going schizophrenic as we speak, then I believe that if case #2 holds, then $\tilde{a}_p$ is the same thing as $a$, hence if there are several prime ideals containing $a$, then all of these prime ideals should contain $\tilde{a}_p$. The author is implying somehow that under certain circumstances, $a$ is not the same thing as $\tilde{a}_p$, and I am completely lost.
Thanks in advance.
After digging a bit through Atiyah-McDonald, I realized that the correspondence theorem between ideals and their extensions in the localization is a correspondence theorem for prime ideals. It does not hold in general for all ideals, and this was my misunderstanding.