split epimorphims and direct sum

Question

Let $M$ and $N$ be modules and $f : M \to N $ a split surjection (that is there exists a map $g : N \to M$ such that $f\circ g=id_N$). Then can we write

$$ M \simeq Ker(f) \oplus Im(f) \ \text{?}$$

Please help me.

Answer 1

Consider the exact sequence $$0\longrightarrow Ker(f)\longrightarrow M\longrightarrow N\longrightarrow0.$$ Note that $(g\circ f)²=(g\circ f)\circ(g\circ f)=g\circ(f\circ g)\circ f=g\circ f$ and $Ker(g\circ f)=Ker(f)$ (because $g$ is injective; check it!). Therefore $g\circ f$ is a projection of $M$. Then $$M=Im(g\circ f)\oplus Ker(g\circ f)=Im(g\circ f)\oplus Ker(f).$$ Since $$Im(f)\simeq\frac{M}{Ker(f)}\simeq Im(g\circ f)$$ the result follows.

Answer 2

Check the splitting lemma over here . https://en.wikipedia.org/wiki/Splitting_lemma here the exact sequence is $0 \rightarrow kerf \rightarrow M \rightarrow N \rightarrow0$