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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.

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Yes, it's true. Suppose that $K+L=N$. Then $K+(L\cap M) = M$, so $L\cap M = M$, since $K\leq_s M$. Can you take it from here?

Here are a couple other results on small modules you can try:

Suppose $K\leq N\leq M$.

  1. $N\leq_s M$ if and only if $K\leq_s M$ and $N/K\leq_s M/K$.
  2. If $H\leq M$, then $H+K\leq_s M$ if and only if $H\leq _s M$ and $K\leq_s M$.
  3. If $K\leq_s M$ and $f\colon M\to N$ is any module homomorphism, then $K\leq_sM$. (The result you asked about is a special case of this one, with $f$ the inclusion of $M$ into $N$).

There is a bit about small and essential submodules in Anderson and Fuller's Rings and Categories of Modules. Also in Lam's Lectures on Modules and Rings.