Dimension of $\mathbb{Q}\otimes_{\mathbb{Z}} \mathbb{Q}$ as a vector space over $\mathbb{Q}$ The following problem was subject of examination that was taken place in June. The document is here. Problem 1 states:

The tensor product $\mathbb{Q}\otimes_{\mathbb Z}\mathbb{Q}$ is a vector space
  over $\mathbb{Q}$ by multiplication in the left factor, i.e.
  $\lambda(x\otimes y)=(\lambda x)\otimes y$ for $\lambda, x,
 y\in\mathbb{Q}$. What is the dimension of
  $\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{Q}$ as a vector space over
  $\mathbb{Q}$?

I only know the definition of tensor product for modules (via universal property). How does one go about calculating dimension of such a vector space? 
Thanks!

There is another solution using category theory : tensor product is the fiber coproduct in the category of commutative rings. $\mathbb{Z} \rightarrow \mathbb{Q}$ is an epimorphism since it's a localization.
If $A \rightarrow B$ is an epimorphism, and $B \rightarrow C, B \rightarrow D$ are any morphisms, then $C \coprod_A D \simeq C \coprod_B D$. Therefore, $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} = \mathbb{Q} \otimes_{\mathbb{Q}} \mathbb{Q}= \mathbb{Q}$
