It has recently been explained to me on this site what direct summands really are. (Here.)
Now I think I have a more difficult question. There is a theorem (easy to prove) which says
For a module $M$ and its endomorphism $f,$ $f$ is a von Neumann regular element of $End(M)$ iff $\ker f$ is a direct summand of $M$ and at the same time $\operatorname{im} f$ is a direct summand of $M.$
I would like to know if there are any useful (or even less useful) necessary and/or sufficient conditions for
$\ker f$ is a direct summand of $M;$
$\operatorname{im} f$ is a direct summand of $M.$
I understand that this is technically open-ended and I can't know exactly what the scope of it is, but based on my (not that great) mathematical experience, I believe this is not a real issue here. However, if I am wrong and this is a very broad subject, please just give me some references and a short outline of the most important facts if it's possible.
I would be especially grateful for a condition that would be equivalent to (a) but didn't use the standard notion of a kernel, but used the notion of the kernel being an equivalence relation (congruence). I am trying to rewrite certain things that are true for modules in the language of semimodules where the module-like kernels simply don't exist.