A noetherian $R$-module $M$ with submodules $L,N$ s.t $L\cap N={0}$ and $M/N\cong L$

Question

With these circumstances, can we conclude that $M=L+N$ ?

All the counter-examples that I know are not noetherian(e.g infinite vector spaces).
And I know that if $M$ has finite length the answer is yes.

Any answer or comment are appreciated.

Answer 1

If you restrict to $N=0$, the question specializes as follows:

Question: Does there exist a Noetherian $R$-module $M$ with a proper isomorphic submodule?

The answer is affirmative, you'll find it already when $R=M={\mathbb Z}$. Let me know if you need more hints.