The problem comes from an exercise of Rings and Categories of Modules.

Let $\left( M\_{\alpha} \right) \_{\alpha \in A}$ be an indexed set of $R$-modules.

The author says $M$ may be isomorphic to the external direct sum $\oplus \_AM\_{\alpha}$, yet not be the internal direct sum of these modules.

I want to find an example.

In my opinion, if A is a finite set or a countable set, the external direct sum and the internal direct sum may be similar.

Maybe my opinion is wrong.

Thanks!