I've got another question. I'm currently studying the section concerning small (inessential or superflous, as you wish) submodules: namely we define $$N\leq_s M$$ if $N\leq M$ and whenever $L\leq M$ is another submodule of $M$ such that $$L+N=M$$ then necessarily $$L=M.$$ One of the first question that came to my mind is: is it true the following
$$K\leq_s M\leq N\Rightarrow K\leq_s N ?$$ I cannot convince myself. If you have references those are welcomed as well.