I'm trying to show the following:
Let $I,J$ be ideals of a commutative ring (with 1) $A$ such that $I_{P}=J_{P}$ for every prime ideal P of R. Here $I_{P}$ means the localization of $I$ in $P$. Then $I=J$.
Well I was thinking in using the following result: let $M$ be an $A$-module. If $M_{P}=0$ for every prime ideal then $M=0$.
But don't we need some kind of assumption like $J \subset I$?
Because if for instance, say $J \subset I$ then $(I/J)_{P} \cong I_{P}/J_{P}$ so $(I/J)_{P}$ is the trivial module for every prime ideal $P$ so $I/J=0$ hence $I=J$.
Is this wrong? How do we proceed?