Somewhere I've read the following:
Theorem Let $I \subset A$ an ideal of a domain $A$. Let $S$ a multiplicatively closed set and let be $I^e$ the image of $I$ in $S^{-1}A$. Let $s \in S$ be such that $I^{ec} \subseteq (I:s)$ then we have $$I=(I:s)\cap (I, s)$$
I never prove it but now I need of this theorem in a demonstration.
Anyone has proof of this fact or can give me some hints?
Thank you
Let $a\in(I:s)\cap(I,s)$. Then $as\in I$ and $a=i+bs$, $i\in I$. We get $bs^2\in I$. It's clear that we have to show $bs\in I$, that is, $b\in (I:s)$ and this holds whether $b\in I^{ec}$. The latter means that there is $u\in S$ such that $bu\in I$, and this happens for $u=s^2$.