I'm trying to find a counterexample to $$ S^{-1} \left(\displaystyle\sum_{i=1}^\infty I_i \right) = \displaystyle\sum_{i=1}^\infty S^{-1}I_i, $$
where the $I_i$ are ideals on a commutative ring $A$ and $S$ is a multiplicatively closed subset of $A$.
The person that suggested this problem to me gave a hint that $\displaystyle\sum_{i=1}^\infty I_i$ should be a large set, but with each $S^{-1}I_i$ small. Could someone give me a hint?
You won't find a counterexample because the equality always holds. Do you know the proof when the sum is finite? The proof in the infinite case is effectively identical.