TITLE: A question about the tensor product of $\mathbb{Q}$
QUESTION [12 upvotes]: I'm reading this blog post about $\mathbb{Q} \otimes_\mathbb{Z} \mathbb{Q}$ and I have two questions about it:

Is a simple tensor a tensor that cannot be written as a sum of tensors?
On the first line, how did they get $\frac{ad}{bd} \otimes \frac{bc}{bd} = \frac{a}{b} d \otimes b \frac{c}{d}$ ?

Many thanks for your help.

REPLY [11 votes]: 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 $\mathbb{Z} \leq N \leq \mathbb{Q}$.