In commutative algebra, a very basic theorem which we study in localizations is the following.
Let $M$ be an $R$-module and $R$ is a commutative ring with unity. Then the following are equivalent:
$M=0$.
The localization $M_{P}=0$ for every prime ideal of $R$.
The localization $M_P=0$ for every maximal ideal of $R$.
The theorem is termed as of type local-global nature; but I didn't understand the philosophical comment.
Q. Is there a simple example illustrating above theorem? (For example, is there an easier equation over a ring, which is difficult to solve within the ring, but easy to solve in the localizations and conclude about solvability of the equation?