I am having many questions about abelian groups, indecomposable groups and the direct product. Here goes :
1) Are all the indecomposable abelian groups (which can't be written as non trivial direct products) known ? I have thought for long that there only was $\mathbb{Q}$, its subgroups, cyclic groups of primary orders and Prüfer groups (complex unit roots of $p^n$ for some $n$, $p$ fixed prime). And then I realised the p-adic integers et their fraction field are indecomposable too. Do you know what is known ? I am having trouble finding things on the matter, I would hope for a "unique decomposition theorem" for all groups that are of some "finite dimension". Does it exist ?
2) I have tried to study the group $\mathbb{Z}^\mathbb{N} / \mathbb{Z}^{(\mathbb{N})}$, it seems interesting (integers sequences modulo integers sequences having finite support) : it does not have any of the indecomposable abelian groups I know (listed above) as direct factors, yet it is not indecomposable (it is its own square). Are there other know groups like this ?
3) A professor suggested to look for subgroups of $\mathbb{Q} \times \mathbb{Q}$, I had trouble finding interesting ones, what do they look like ?
4) I have that intuition that indecomposable groups show be at most countable, is that true ?
5 ) Are there good invariants to study abelian groups (I mainly use n-divisibility of its elements, and their order) ?
6 ) I had this idea to study groups like subgroups of $\mathbb{Q} \times \mathbb{Q}$ : any at most countable abelian group can be written as a directed colimit of finitely generated groups thus as a colimit of a diagram of well know groups and matrices between them : could this approach work to study abelian groups ?
7) Is it true that for every abelian group $G$ and $H$, for every subgroup of $K$ of $G \times H$, there are two subgroups $G'$ of $G$, $H'$of $H$, such that $K$ is isomorphic to $G' \times H'$. It seems wrong, I found a counter example in the non abelian case, not in the abelian case.
8) Are some of these questions studied for $R$-modules in general ? What are know indecomposable $R$-modules etc ?
Sorry for all the questions,
Thank you in advance,
Ludovic