So we've proven in class that when the sum of two sets, say vector spaces, is such that each element can be uniquely expressed as the sum of two elements in each set, then it is an Internal Direct Sum e.g. \begin{equation} \mathbb{R}^2=\mathbb{R}e_1 \oplus \mathbb{R}e_2 \end{equation} for a trivial example.
Then we were able to prove that when the Internal Direct Sum "applies", like the example above, then the External Direct Sum is isomorphic to the Internal Direct Sum. Hence, the two are in some sense "interchangeable" and we should use the "form" which makes the problem "easier".
In the case of modules, if we must show that \begin{equation*} \mathbb{Z}/6\mathbb{Z}\simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z} \end{equation*} this would be the External Direct Sum, right? Because the above doesn't seem to make sense as an Internal Direct Sum.
Does this mean if the context doesn't make sense for an Internal Direct Sum, I should assume the External Direct Sum?
Lastly, If I needed to prove that the first example was an Internal Direct Sum, would it be sufficient to show that it is an External Direct Sum?
Any clarification on my confusion above is much appreciated.