A question about the tensor product of $\mathbb{Q}$

A simple tensor is an element of a tensor product that can be written in the form x⊗y. It could also be written as a sum of tensors, but not all sums of tensors can be written as a single x⊗y (in general).

The chain of equalities in the blog post was incorrect. They are not equal. Here is a similar but correct chain of equalities you might find useful:

$$ \frac{a}{b} \otimes \frac{c}{d} = \frac{ad}{bd} \otimes \frac{c}{d} = \frac{a}{bd} d \otimes c \frac{1}{d} = \frac{a}{bd} c \otimes d \frac{1}{d} = \frac{ac}{bd} \otimes 1$$

I would be hesitant to write $\frac{ac}{bd}(1\otimes 1)$ unless you are considering $\mathbb{Q}$ as a left-$\mathbb{Q}$, right-$\mathbb{Z}$ module.

At any rate, the map $q \mapsto q \otimes 1$ is an isomorphism from $\mathbb{Q}$ to $\mathbb{Q}\otimes \mathbb{Q}$, or indeed from $\mathbb{Q}$ to $\mathbb{Q} \otimes N$ for any abelian group $\mathbb{Z} \leq N \leq \mathbb{Q}$.