Let $M = K + L$ and let $f: M \to N$ be an epimorphism. Prove that $N = f(K) \oplus f(L)$ if $K \cap L = \operatorname{Ker}f$

Question

Let $M = K + L$ and let $f: M \to N$ be an epimorphism. Prove that $N = f(K) \oplus f(L)$ if $K \cap L = \operatorname{Ker}f$

I think we should start with take an element from $K + L$ and then we can use $f(M)=f(K + L)=N$ but I don't know how to do the rest.

The book is rings and categories of modules. Writers Anderson and Fuller

Section 5 exercise 5

Answer 1

Clearly, $f(K)+f(L) = f(K+L)= N$.

Let $x\in f(K)\cap f(L)$. Then there exist $k\in K$ and $l\in L$ such that $f(k)=x=f(l)$, and hence $f(k-l)=0$. So $k-l\in\ker(f) = K\cap L\subset L$, implying that $k = (k-l) + l \in L+L=L$. This means that $k\in K\cap L = \ker(f)$ and hence $x=f(k)=0$. So $f(K) \cap f(L) = \{0\}$.