Let $N$ and $K$ be submodules of an $R$-module $M$, with $Ann(N)$ and $Ann(K)$. Give an example where $Ann(N) +Ann(K) \ne Ann(N\cap K)$
My attempt :
We consider $M=\Bbb R^2$ as an $\Bbb R$-module. Let , $N= \Bbb Re_1,K=\Bbb Re_2$.
Then, $Ann(N)=0=Ann(K)$ but , $Ann(N\cap K)=Ann(\Bbb Re_1 \cap \Bbb Re_2)=Ann((0,0))= \Bbb R$
Does my example work? Please point out mistakes, if any.