I am confused as to whether there is a module that is not completely decomposable. If the module $M$ has finite length, this is easy: if $M$ isn't indecomposable, $M$ can be written as $M_1\oplus M_2$; then have a similar discussion on $M_1,M_2$; this process will terminate in a finite number of steps. But in general, I think maybe we can't do that as before. For example, we can't use this to show that the real number $\mathbb{R}$ as a vector space over $\mathbb{Q}$ can be written as a direct sum of a family of $\mathbb{Q}$, which is the rational numbers.
So, how can we prove that every module is completely decomposable, or there exists a module which is not completely decomposable?
