TITLE: $M/\mathfrak{m}M\simeq A/\mathfrak{m}$, for any maximal ideal $\mathfrak{m}\subset A$
QUESTION [5 upvotes]: Let $M$ be an $A$-module. Is it true that if $M/\mathfrak{m}M\simeq A/\mathfrak{m}$, for any maximal ideal $\mathfrak{m}\subset A$, then $ M\simeq A$?

REPLY [5 votes]: Let $A$ be a Dedekind domain, of class number greater than $1$, and let
$I$ be a non-principal ideal of $A$. Then $I/\mathfrak{m}I\cong
A/\mathfrak{m}$ for all $\mathfrak m$ but $A\not\cong I$ as $A$-modules.