$M$ is isomorphic to the external direct sum, is not the internal direct sum of submodules. 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!
 A: I assume that the $M_\alpha$ are implicitly submodules of $M$. Take $M = k^2$ for $k$ your favorite field. Let $M_1= M_2 =\{(x,0):x\in k\}$, i.e. the $x$-axis. The internal sum of $M_1$ with $M_2$ is not $M$, while $M\cong M_1\oplus M_2$ as $k$-vector spaces, since they are both of dimension $2$.
A: There are two notions of “direct sum”:

*

*If $\{M_i\}_{i \in I}$ is a family of $R$-modules, we define the external direct sum of $\{M_i\}_{i \in I}$ to be
$$
\bigoplus_{i \in I} M_i := \left\{ (m_i)_{i \in I} \in \prod _{i \in I} M_i : |\{i \in I : m_i \neq 0\}| < \aleph_0 \right\}.
$$
That is, $\bigoplus_{i \in I} M_i$ is the set of all elements of $\prod _{i \in I} M_i$ which have only finitely many non-zero coordinates.
It is easy to verify that $\bigoplus_{i \in I} M_i$ is a submodule of $\prod _{i \in I} M_i$, so it is an $R$-module, naturally.

*If $\{M_i\}_{i \in I}$ is a family of submodules of a given $R$-module $M$, we say that $M$ is the internal direct sum of $\{M_i\}_{i \in I}$ if the map
$$
\bigoplus_{i \in I} M_i \to M, \quad (m_i)_{i \in I} \mapsto \sum_{i \in I} m_i
$$
is an isomorphism. That is to say, if each element of $M$ can be written uniquely as a (finite) sum of the form $\sum_{i \in I} m_i$, with $(m_i)_{i \in I} \in \bigoplus_{i \in I} M_i$. Unfortunately, this situation is written as $M = \bigoplus_{i \in I} M_i$, although $M$ is not even a subset of $\prod _{i \in I} M_i$.

Thus, as you can see, we can form the external direct sum for every family of modules, but we only have internal direct sums for families of submodules of a module if certains conditions holds.
One thing is obviously true (by definition): if a module is the internal direct sum of some family of submodules, then it is isomorphic to the external direct sum of such submodules.
So, now, what the author said is may be easier to see.
