The difference between $\prod$, $\coprod$, $\bigoplus$, and $\sum$ (especially in the category **R-Mod**)

Question

I am interested in knowing what the difference between products, coproducts, direct sums, and internal direct sums are in a general category as well as in the category of R-Modules. Different sources have subtly different notations for these concepts which is contributing to my confusion. I know that in R-Mod there are various theorems saying that some of these concepts are equivalent but from Hungerford I am not always sure what he is talking about when he states these theorems.

Answer 1

I'm going to assume these are indexed by sets. They can be indexed by other things but then their meaning changes, though often in a way that is closely related to the set-indexed case.

As operations on objects of a (possibly implicit) category, $\prod$ usually means categorical product, i.e. the limit of the discrete diagram generated from the indexing set. Similarly, $\coprod$ and $\sum$ usually mean categorical coproduct. $\bigoplus$ is usually more commonly used with additive categories where it is usually taken to mean the coproduct. However, for finite indexing sets coproducts and products coincide and are called biproducts. So restricted to finite indexing sets the $\bigoplus$ notation emphasizes this biproduct nature. However, once you go beyond finite indexing sets coproducts and products usually diverge, and, again, $\bigoplus$ is usually taken to be the coproduct in that case.

R-Mod is an additive category (in fact, the representative one), and so $\bigoplus$ is often used with $\prod$. With no explicit definitions otherwise, $\bigoplus$, $\coprod$, and $\sum$ would all be synonyms. Nevertheless, it may be useful to distinguish $\coprod$ (or $\sum$, they aren't usually used together) from $\bigoplus$ so you can establish the theorem that they are the same. Either way, the category of rings, say, is not additive and so finite coproducts are not, in general, biproducts and thus it's sensible to use $\coprod$ or $\sum$ for rings and $\bigoplus$ for the modules. This allows you to quickly recognize which rules and properties hold just by looking at the syntax.