Consider $p$ is a prime, the principal ideal of $\mathbb{Z}$ generated by $p$ denoted by $(p)$, the direct product $\prod_{p}\mathbb{Z}/(p)$ and the direct sum $\bigoplus_{p}\mathbb{Z}/(p)$ run over all primes.
Find an injective $\mathbb{Z}$-module homomorphism from $\mathbb{Q}$ to $\mathscr{A}$, where $$\mathscr{A}=\prod_{p}\mathbb{Z}/(p)/\bigoplus_{p}\mathbb{Z}/(p),$$ that is, a quotient $\mathbb{Z}$-module.
Define $\varphi:\mathbb Q\to\mathscr A$ for integer $n$ as $$\varphi(n)= [[n]_2,[n]_3,[n]_5,\ldots].$$ Then note $\varphi(n)$ is invertible in $\mathscr A$. Indeed, each $\mathbb Z/(p)$ is a field, so we can define the $p$-th component of $\varphi\left (\frac 1n\right)$ to $[n^{-1}]_p$ if $p\nmid n$, and $0$ otherwise.
It's trivially checked that $\varphi$ is an injective $\mathbb Z$-module homomorphism.
Remark. Actually $\bigoplus_{p}\mathbb{Z}/(p)$ is the torsion subgroup of $\prod_{p}\mathbb{Z}/(p)$, so $\mathscr A$ is torsion-free and divisible abelian group, and by well-known theorem it is a vector space over $\mathbb Q$.