Is $\Bbb Q$ a decomposable module over $\Bbb Z$ or not? 
Is $\Bbb Q$ a decomposable module over $\Bbb Z$ or not?

My attempt:
let, $p_1,p_2,\dots,p_k,\dots$ be an enumeration of primes in $\Bbb N$. Then, can't we write  $\Bbb Q = \Bbb Z \oplus Z(\frac{1}{p_1})\oplus \Bbb Z(\frac{1}{p_2})\oplus \dots \oplus \Bbb Z(\frac{1}{p_k}) \oplus \dots $ ?
If the above expression is true, for some fixed $l \in \Bbb N$, we consider $A=\Bbb Z \oplus Z(\frac{1}{p_1})\oplus \Bbb Z(\frac{1}{p_2})\oplus \dots \oplus \Bbb Z(\frac{1}{p_l})$ and $B=Z(\frac{1}{p_{l+1}}) \oplus \dots $ .
Then we are able to write $\Bbb Q$ as a direct sum of two non-zero  $\Bbb Z$-modules.
Please give a correction if I'm mistaken. Thanks in Advance for help!

Suppose $\mathbb{Q}$ is decomposable as $\mathbb{Q}=X\oplus Y$. Then $X$ is 


*

*divisible, because it is a homomorphic image of $\mathbb{Q}$;

*torsionfree, because it is a subgroup of $\mathbb{Q}$.


Similarly for $Y$. Therefore $X$ and $Y$ are vector spaces over $\mathbb{Q}$ and $\mathbb{Q}=X\oplus Y$ is a decomposition as direct sum of subspaces. Hence either $X$ or $Y$ must have dimension $0$.
