When flat submodule is direct summand?

Let $M$ be a $\mathbb{Z}$-module. And $N \subset M$ is a submodule. Assume that $N$ is flat as a $\mathbb{Z}$-module. Then I'm wondering if $$M/N \subset M \Rightarrow N \ is \ direct \ summand \ of M$$ is true. But $M/N \subset M$ means that there is an injective group homomorphism $M/N \rightarrow M$.

Is there a counter example?

• What does $M/N\subset M$ mean? – Lord Shark the Unknown Apr 14 '18 at 6:02
• What do you mean by "an inclusion"? – Eric Wofsey Apr 14 '18 at 6:21

Let $$M=\Bbb Q\oplus\bigoplus_{n=1}^\infty(\Bbb Q/\Bbb Z)$$ and let $N$ be the $\Bbb Z$ embedded in the $\Bbb Q$ factor. Of course $N$ is flat, and $M/N\cong \bigoplus_{n=1}^\infty(\Bbb Q/\Bbb Z)$ but $N$ is not a direct summand of $M$.