An $R$-module $M$ is said to be a semisimple module if it is a direct sum of its simple submodules.
I've proved the following equivalent characterization for semisimple modules:
$M$ is semisimple iff every submodules of $M$ is a direct summand.
From the above proposition we can easily show that every submodule and factor module of a semisimple module $M$ are alosa semisimple.But the converse is not always true.
I want to find a module $M$ whose proper submodules and nonzero factor modules are semisimple but $M$ is not semisimple.