1
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ What does $M/N\subset M$ mean? $\endgroup$ – Lord Shark the Unknown Apr 14 '18 at 6:02
  • 1
    $\begingroup$ What do you mean by "an inclusion"? $\endgroup$ – Eric Wofsey Apr 14 '18 at 6:21
2
$\begingroup$

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 $M$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.