Can I Find A Map from a Module M to the kernel of a map p from M to M?

Question

I have a module homomorphism $p:M\rightarrow M$. I would like to find another module homomorphism $\phi:M\rightarrow \ker(p)$. Finding such a thing seems to be very challenging however. Is this possible? Note also that $p^2=p$.

Answer 1

Let $\phi=1_M-p$. Then $p\circ \phi=p(1_M-p)=p-p^2=p-p=0$, so $\phi:M\to\ker p$. But there are many such maps $\phi$; taking $\phi$ to be the $0$ map would also define a map from $M$ to $\ker p$.