$R$ -module homomorphism

Question

Let $M,N$ be $R$-modules. Suppose there exists $R$-module homomorphism $\phi:M \to N$ and $\psi:N \to M$ such that $(\psi \ \circ \ \phi)(m) =m, \forall m \in M. $ Could anyone advise me on how to prove $N \subseteq \text{ker}(\psi) \oplus \phi(M) \ ?$

Hints will suffice, thank you.

Answer 1

Hint: write $n= \left(n-\phi \circ \psi \left(n\right) \right) + \phi \circ \psi \left(n\right)$