Here is a question I think should be elementary but I am having trouble thinking about it.
Let $A$ be a commutative, unital ring, and let $I,J,K$ be ideals of $A$.
We have an inclusion $I\cap K + J \subset (I + J)\cap (K+J)$, which in general can be proper, though they will have the same radical.
On the other hand, if $I\subset K$, then $I\cap K + J = I + J = (I+J)\cap (K+J)$.
What if in place of $I,K$ we have a countable descending sequence of ideals $I_1\supset I_2\supset I_3\supset\dots$?
Does $J + \bigcap I_i = \bigcap (J + I_i)$?
Another way to ask the question is this: does taking the intersection of a descending sequence of ideals $(I_i)$ commute with taking the quotient by an arbitrary ideal $J$? I.e. does the image of $\bigcap I_i$ in $A/J$ equal the intersection in $A/J$ of the images of each $I_i$?
If it's true, what's the proof? If it's not true, is it true under some reasonable hypotheses (e.g. $A$ is noetherian)?