I am having trouble proving the following statement.
Let $A$ and $B$ be infinite, disjoint sets of prime numbers with $5 \notin A\cup B$. Set $e_2' = 5e_1+2e_2$ where $e_1,e_2,e_3$ denotes the canonical basis of $\mathbb{Q}^3$. Consider the two $\mathbb{Z}$-submodules $N,N'$ of $\mathbb{Q}^3$ given by $N = \sum_{p \in A}\mathbb{Z}\frac{e_2}{p} + \sum_{q \in B}\mathbb{Z}\frac{e_3}{q}+\mathbb{Z}\frac{e_2+e_3}{5}$ and $N' = \sum_{p \in A}\mathbb{Z}\frac{e_2'}{p} + \sum_{q \in B}\mathbb{Z}\frac{e_3}{q}+\mathbb{Z}\frac{3e_2'+e_3}{5}$. Show that $N$ and $N'$ are indecomposable and that $N \ncong N'$.
All the theorems I know of speak of finitely generated modules which these aren't. For the indecomposable part I have tried showing that any two submodules of $N$ (and $N'$) have a nonempty intersection but this seem to not work because of the two nonzero coordinates.
I have tried assuming an isomorphism of modules but again I am having troubles because the submodules are not finitely generated.
Any help is greatly appreciated.
