Are operations on ideals and direct sums/products of abelian groups related?

Question

References will suffice for answers, these are dumb questions and I don't want to waste too much of anyone's time with them. So will short answers.

1. Is it true that the sum of ideals is isomorphic to their direct sum (when considered as abelian groups) if and only if their intersection is 0?

Seemingly this would generalize the situation from linear algebra.

2. I know that the direct sum and product of abelian groups coincide for finite index sets, but not for infinite index sets.

Since the sum and product of finitely many ideals do not coincide, does the answer to 1 above imply that there is no relationship between the product of ideals and the direct product of ideals (considered as abelian groups)?

3. Do the sum operation for ideals and the direct sum operation for abelian groups have similar names because both operations only involve finitely many non-zero terms for infinite index sets?

4. Building off of the answers to 2 and 3, is the similarity of the names of product of ideals and direct products of abelian groups only a coincidence? Or is it meant to reflect some relationship between them?

(I.e. because the former is intended to refer to a specific defined ring multiplication operation, whereas product in the latter is just meant to refer to some dual or adjoint notion to the categorical notion of direct sum?)

In particular, is there or is there not any relationship between the two facts that (i) only the product of finitely many ideals is defined, and (ii) the direct product and sum only coincide for finite index sets?

Answer 1

1. Yes. In general, $I+J\simeq (I \oplus J)/\{(x, -x) \;|\; x \in I\cap J\}$ as $R$-modules.

2. The direct product does not seem to be directly related to product of ideals. The closer thing to doing this is the tensor product - namely, tere is always an epimorphism $I \otimes_R J \twoheadrightarrow IJ$. I think it is not injective in general, although I am not sure now.

3. The connection is the same as in 1. More precisely, one has always an epi $\bigoplus_j I_j \rightarrow \sum_j I_j$ and the kernel is spanned by tuples of elements that sum up to $0$.

4. Hard to say. The product of ideals comes from the product of elements, whereas the product of $R$-modules is named after the cartesian product (of sets), so in that sense I guess yes.