About the definitions of direct product and direct sum of modules.

Question

Direct product of modules can contain infinitely nonzero elements, but direct sum of modules must contain finitely nonzero elements. What's the point of the definitions? Why the definitions are defined like these ?

Answer 1

Additionally, you will see in category theory (if you go down that road) that direct products and direct sums satisfy different universal mapping properties: https://en.wikipedia.org/wiki/Direct_sum_of_modules

As you notice in the Wiki page, the direct sum of modules is a $coproduct$, meaning it satisfies the universal mapping property $opposite$ of that of the direct product of modules; the direct product satisfies the universal mapping property of a $product$ of objects in a category. If you're a bit mixed up by that, don't worry too much. Suffice to say that constructions in category theory are often identified and delineated by their universal mapping properties, so products are unique up to isomorphism in a given category. Coproducts, too, are unique up to isomorphism by a duality argument. And it is by their universal mapping properties that products (e.g. direct products of modules) and coproducts (e.g. direct sums of modules or disjoint unions mod some equivalence relation in other familiar categories) are most clearly distinguished (for me).

For an introduction to category theory (which should be appropriate for you given your experience with modules so far), you can look at Vakil's Algebraic Geometry (using a quick Google search with the word "PDF" at the end) or Steve Awodey's Category Theory text (using the same kind of search). Awodey also gave some lectures at Oregon State on the foundations of category theory which you can find conveniently on YouTube by just looking up his name.

Answer 2

There are many differences between the two, for example you can prove yourself that the ring $\prod_{n\geq 1}\mathbb{Z}_{p^n}$ is not Noetherian. Since each $\mathbb{Z}_{p^n}$ is finite it is trivially Noetherian, we see that the property of being Noetherian is not preserved under direct products whereas it is preserved under direct sums.

In some sense, direct sums are smaller than direct products and hence more finiteness conditions are preserved under direct sums.