All Questions

Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)? Edited to add: As no answers are forthcoming, does anyone ...

1) Is there any characterization of $\Bbb{Z}$-modules with finite uniform dimensions? 2) Find two $\Bbb{Z}$-modules $N, M$ such that $N\leq M$ and $M\hookrightarrow N$ but $N$ is not isomorphic to $M$...

I am trying to prove that for any divisible torsion $\mathbb{Z}$-module $V$, this map $$f:\mathbb{Q}/\mathbb{Z}\otimes_E\text{Hom}(\mathbb{Q}/\mathbb{Z},V)\longrightarrow V\mbox{ defined by } f((q+\...