Kernel of projective module

Question

Suppose that $f \colon M \to N$ is a homomorphism of R-modules such that M, N and N/im(f) are projective. Is is true that ker(f) is projective?

Thanks a lot for any help!

Answer 1

If $N$ and $N/\operatorname{Im}f$ are projective, the short exact sequence $$0\longrightarrow \operatorname{Im}f\longrightarrow N \longrightarrow N/\operatorname{Im}f\longrightarrow 0 $$ splits, so $\operatorname{Im}f$ is a direct summand of the projective module $N$ and therefore is projective.

Can you prove similarly that $\ker f$ is projective?