Example of non distributive module

Question

I'm looking for a module that has three submodules $P,Q,R$ such that $P\cap (Q+R) \neq P\cap Q +P\cap R $ I'm struggling to find this example because I'm not that familiarized in module theory Can You help me? Thanks

Answer 1

Take a field $k$, and the vector space $k^2$. Let $P=\{(x,x):x\in k\}$, $Q=\{(x,0):x\in k\}$, and $R=\{(0,y):y\in k\}$. Then $Q+R=k^2$, so $P\cap(Q+R)=P$, but $P\cap Q=0=P\cap R$.

Answer 2

The smallest example is to take the Klein $4$-group viewed as a $\mathbb{Z}$-module. Then take $P$, $Q$, and $R$ to be the nontrivial proper subgroups, any two of which generate the whole group, but have trivial intersection.

Note that in general you can consider abelian groups to be $\mathbb{Z}$-modules, so you can use them to provide examples/counterexamples for general statements about modules, if you are more familiar with them and modules in general.