I would like to pose a question which is, from my point of view, surprisingly elementary, difficult and interesting at the same time. I hope that somebody could enlighten me.
Let $k$ be a field and $A$ be a $k$-algebra which is a domain, with fraction field $K$. Let $L_1, \dots L_d$ be a finite number of elements of $K$. Let $A[L] \subseteq K$ be the $A-$algebra generated by the elements $L_1, \dots , L_d$, inside $K$.
Consider the localisation functor $(-)^*$, from the category of $A$-modules to the category of $K$ modules, sending $M$ to $M^*= (A \setminus \{0 \})^{-1} M$. We have a canonical map of $A$-modules $\iota : M \longrightarrow M^*$.
Then, consider the functor $(-)[L]$, from the category of $A$-modules to the category of $A[L]$-modules, sending $M$ to the $A[L]$-module generated by $\iota(M)$, inside $M^*$.
The question is: is the functor $(-)[L]$ exact (or left exact or right exact)? If not, is it exact (or left exact or right exact) under some condition? Or else, can you find a condition on an exact sequence of $A$-modules $0 \longrightarrow M \longrightarrow N \longrightarrow K \longrightarrow 0$ (resp. $0 \longrightarrow M \longrightarrow N \longrightarrow K $) such that applying $(-)[L]$ preserves the exacteness of this sequence.
Of course, the problem is easy if $L_i= \frac{1}{a_i}$ for some $a_i \in A$, for any $i$, but if this is not the case, then in my opinion the problem becomes quite hard. Also, if I'm not wrong, the functor $(-)[L]$ should preserve injections and surjections. But exactness in the middle of a sequence is a mess. I would be glad to have some help!