Does every mon-small submodule contain a nonzero supplement submodule of the module.

Question

Definition: Let $M$ be an $R$-module. Then

1. A submodule $N$ is called small (in $M$) if $N+K=M\implies K=M,\forall K\leq M$.
2. A submodule $N$ is called a supplement of a submodule $K$ (in $M$) if $N$ is minimal with respect to the property $N+K=M$.
3. A submodule $N$ is called a supplement submodule if it is a supplement of some submodule $K$.

My Question: Does every mon-small submodule contain a nonzero supplement submodule of the module.

Answer 1

False.

You can take example of $\mathbb{Z}$-module $\mathbb{Z}$. Its every nonzero submodule is nonsmall while it has no nonzero supplement submodule.