Module Decomposition into external sum of Quotient and Submodule

Question

I am unsure whether for a Module $M$ and a submodule $N \subseteq M$ we have that $$ M \cong N \oplus_{ext} M/N $$ It seems somewhat intuitive that $ N \oplus_{ext} M/N $ contains all information to describe an element of M.

But I haven't been able to find a suitable isomorphism. In particular because there seems to be no good choice of inverse of the canonical projection $q: M \twoheadrightarrow M/N$.

The more I look at it the more I get the feeling that this is too good to be true...

Answer 1

It is actually too good to be true:

Let $ M = \Bbb Z$ be the Module over itself and $N = 2\Bbb Z$ the submodule. Then for $(0,1)$ in $ 2\Bbb Z \oplus \Bbb Z/2\Bbb Z$ we have $2(0,1) = 0$.

But since $\Bbb Z$ is torsion-free we cannot have $\Bbb Z \cong 2\Bbb Z \oplus \Bbb Z/2\Bbb Z$.