fully faithful inclusion of Q-vector spaces into Ab

Let \(\varphi: V \rightarrow W\) be a group-homomorphism between \(\mathbb{Q}\)-vector spaces.
So let \(\frac{p}{q} \in \mathbb{Q} \setminus \{0\}\) be a scalar and \(v \in V\).
Then we want to show that

\begin{align*} \varphi( \frac{p}{q} v) = \frac{p}{q} \varphi(v) \end{align*}

We know that \(\frac{p}{q} = \sum_{i=1}^p \frac{1}{q}\) and hence

\begin{align*} \varphi( \frac{p}{q} v) =& \varphi( \sum_{i=1}^p \frac{1}{q} v) \\ =& \sum_{i=1}^p \varphi(\frac{1}{q} v) \\ =& p \cdot \varphi( \frac{1}{q} v) \end{align*}

Now \(V,W\) are in particular uniquely divisible abelian group, so \(\frac{1}{q} v\) is characterized by the property that \(\sum_{i=1}^q \frac{1}{q} v = v\).
Analogously \(\frac{1}{q} \varphi(v)\) is characterized by \(\sum_{i=1}^q \frac{1}{q} \varphi(v) = \varphi(v)\).

Now this shows

rational vector space is a property

The question whether the restriction of scalars functor is fully faithful
see:

• restriction of scalars for a localization is fully faithful
  

use:

• heart of spectra is Ab
  
• heart of the t-structure of modules over a connective ring spectrum (todo: \(\mathrm{ModSp}(\mathbb{Q})^{\heartsuit} \simeq \mathrm{ModAb}(\mathbb{Q}) \simeq \mathrm{Vect}_{\mathbb{Q}}\)
  
• rational module spectra is a full subcategory of Spectra