Direct sum and modules

Question

Let $M$ be an $R$-module and $N$ be a submodule of $M$. Can we always write $M \cong N \oplus M/N$ as $R$-modules ? If not, then under what conditions on $M$, the direct sum holds ?

Answer 1

(This question has almost certainly been asked before, but it seemed easier to just type this up than search for a previous answer).

There is always an exact sequence

$$0 \to N \to M \to M/N \to 0$$

I will state when this sequence is split exact (as pointed out in the comments, this is not the only possibility for which $M \cong N \oplus M/N$). Some cases when it is split:

i) $M/N$ is projective
ii) $N$ is injective

Beyond these two cases, there is not much more that can be said in general (edit: as Pete L. Clark mentions, these are both special cases of $\operatorname{Ext}^1(M/N,N) = 0$). Notice that this is not really a condition on $M$, but rather on the embedding $N \hookrightarrow M$ (or the quotient $M/N$).

Answer 2

No, it is not true. Consider the cyclic of order 4. It is a Z-module. Let N be the subgroup of order 2, then you get a counter example.