Not every submodule has a complement submodule Precisely, suppose $S$ is a submodule of the $R$-module $M$, then it is not necessarily possible to decompose $M$ as 
$$ M = S \oplus  S^c . $$
Any example?
It seems that there are many differences between vector spaces and modules. But I still cannot see the reason behind.
 A: Finding a direct-sum complement can fail in at least two ways:


*

*$S$ could be an essential submodule (this means that for another submodule $T$, $S\cap T=\{0\}$ implies $T=\{0\}$)

*$S$ could be a superfluous submodule (this means that for another submodule $T$, $S+T=M$ implies $T=M$.)
A good place to look for counterexamples is in rings which only have trivial idempotents, since idempotents correspond to module decompositions of the ring as a module over itself.
If you consider $\Bbb Z/4\Bbb Z$ as a module over itself, this module has exactly three submodules arranged in a line. The ring has no nontrivial idempotents. It's easy to see that the nontrivial submodule $2\Bbb Z/4\Bbb Z$ is both essential and superfluous at the same time, and so it (doubly) can't have a direct-summand complement.
Semisimple rings are exactly the class of rings for which you can always find direct-summand complements.

It seems that there are many differences between vector spaces and modules. 

Of course. You might be interested in this question: Pathologies in module theory 
