When quotient module become a direct summand?

Question

let $M$ be a $R$- module and $N$ be a submodule of $N$. If $L$ be another module such that $L \cong M/N$ when we could say $L \cong M \oplus N$.? What if we put $N = ker\varphi$ where $\varphi : M \rightarrow L$.

Answer 1

This is the same thing as asking

When does the exact sequence $0\to N\to M\to L\to 0$ split?

The splitting lemma is the standard result in basic homological algebra that characterizes when this happens. I'm not sure you're going to get a better answer than that.

There are a a few standard types of modules which are relevant to the question of splitting, though. Namely, if $N$ is an injective module or if $L$ is a projective module then the sequence is guaranteed to split.

Also if $M$ is a semisimple module, then the sequence must split. This can happen completely independently of injectivity and projectivity. For example, if you take a ring with a simple module $S_1$ that isn't injective and a simple module $S_2$ which isn't projective, then $0\to S_1\to S_1\oplus S_2\to S_2\to 0$ splits (obviously) but it certainly doesn't do so because of the reason of projectivity or injectivity mentioned above.