Here's the exercise:
Show that if $R=\mathbb{Z}$, $I=\mathbb{Z}^{+}$, and $M_{i}$ is the cyclic group of order $i$ for each $i\in I$, then the direct sum of the $M_{i}$'s is not isomorphic to their direct product. [Look at torsion.]
I am confused about the hint to look at torsion (which I am assuming means something like show that the corresponding torsion submodules are not isomoprhic). Could I simply argue that they are not isomorphic as groups since one contains elements of infinite order (e.g. $(x_{1},x_{2},...)$ where $x_{i}$ is a generator for each $M_{i}$) and one doesn't?
Disclaimer: I am only looking for specific answers to any of my questions and not any worked out solutions.
In my opinion, you've already solved the exercise; and moreover, you have solved it using the given hint! To distinguish $\bigoplus_{i\in I}M_i$ and $\prod_{i\in I}M_i$, you looked at the torsion subgroups of these two groups, and you determined that the former is equal to its torsion subgroup (because, as you said, it has no elements of infinite order) and the latter isn't equal to its torsion subgroup (because you've produced an element of infinite order in it).