Let $M$ and $N$ be $R$ modules ($R$ commutative with identity). Is it true that if fot for every prime ideal $P$, $M_P \cong N_P$ (as $R_P$ modules) then $M \cong N$ ? Clearly the question is true if $M$ or $N$ is zero. But what about the non-zero case !?
Let $M$ and $N$ be $R$ modules ($R$ commutative with identity). Is it true that if fot every prime ideal $P$, $M_P \cong N_P$ (as $R_P$ modules) then $M \cong N$ ? Clearly the question is true if $M$ or $N$ is zero. But what about the non-zero case !?