Given a commutative unital ring $R$, and a multiplicative subset $S\subseteq R$, I know that for two $R$-submodules $M_1,M_2$ of $M$, we have:
(I) $S^{-1}M_1 \cap S^{-1}M_2= S^{-1}(M_1\cap M_2)$
(II) $S^{-1}M_1+S^{-1}M_2=S^{-1}(M_1+M_2)$
My question is does this also hold for infinite intersections and sums. I am pretty sure that a sum of localizations is equal to the localization of the sum, simply because every element in a sum of modules can be discussed within a finite sum. I suspect that this is untrue for the intersection, but I could not think of a counter-example.
As suggested by user26857 the counter example for the question is:
For the $\mathbb{Z}$-modules $M_n=n\mathbb{Z}$, and the multiplicative set $S=\mathbb{Z}\setminus\{ 0\}$ we have:
$S^{-1}\big( M_n \big)=\mathbb{Q}$, $\quad \underset{n=1}{\overset{\infty}{\bigcap}} M_n=(0) \quad$ and $\quad S^{-1}\big( 0 \big)=(0)\subseteq \mathbb{Q}$.
Therefore:
$\underset{n=1}{\overset{\infty}{\bigcap}} S^{-1}M_n=\underset{n=1}{\overset{\infty}{\bigcap}}\mathbb{Q}=\mathbb{Q}\neq (0)=S^{-1}\Big( \underset{n=1}{\overset{\infty}{\bigcap}}M_n \Big)$