If $N \le M$ , $M$ finitely generated , $N$ is isomorphic to a direct summand of $M$ , then is $N$ a direct summand of $M$

Question

Let $N$ be a submodule of a finitely generated module $M$ such that $N \cong P$, where $M=P \oplus Q $ for some submodule $Q$ of $M$. Is it true that there is a submodule $N'$ of $M$ such that $M =N \oplus N'$ ?

I have a feeling this is not true , but I am not sure. Please help

Answer 1

Hint: Consider the special case $R=M={\mathbb Z}$ and $Q=0$.