counter example in $R$-modules law

Question

Let $M$ be an $R$-module and $K$, $L$ and $N$ submodules. I would like to find a counterexample to the equality

$$N\cap (K+L) =(N\cap K)+(N\cap L)$$

I can prove the equality is true when $K \subset N$. I.e., what $N,K,L$ exist such that ...

Answer 1

Try three lines through the origin in $\mathbf{R}^2$.