Let $R$ be a commutative ring and let $Q$ be an $R$-module. Let $M, N, P$ be submodules of $Q$ without satisfying any additional requirements. Let $M+N$ denote the linear sum of submodules.
What can be said about the sum of intersections $$(M\bigcap P) + (N\bigcap P)?$$
Is it contained in the intersection $(M+N)\bigcap P$, that is, is it true that
$$(M+N)\bigcap P\supseteq (M\bigcap P) + (N\bigcap P) ?$$
Or is it the other way around, that is,
$$(M+N)\bigcap P\subseteq (M\bigcap P) + (N\bigcap P)?$$
Or is it undecided, meaning it depends on the concrete case and hence, there is no generally valid inclusion?
This is asking if the lattice of submodules of $Q$ is a distributive lattice or not. A module whose lattice of submodules is a distributive lattice is called a distributive module.
If $Q$ is distributive, then the identity holds all the time. If it is not distributive, then there are submodules $M,N,P$ which fail the condition.
That $(M+N)\bigcap P\supseteq (M\bigcap P) + (N\bigcap P)$ is trivial. If $x+y\in (M\bigcap P) + (N\bigcap P)$, then $x+y\in M+N$ and $x+y\in P+P=P$, so $x+y$ is in their intersection. But the reverse inclusion is not always true.
In fact, every nonzero (with identity) ring has a nondistributive module.
There are examples, though, of rings whose regular module $R_R$ is distributive. $\mathbb Z$ is such an example.