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Vell Cool Sand. Some file need OPm.Q:
Isomorphism $ \Bbb Q \otimes_{\Bbb Z} \Bbb Q \rightarrow \Bbb Q \otimes_{\Bbb Z} \Bbb Q$
Let us denote $\Bbb Z$-modules $\Bbb Q \otimes \Bbb Q $ and $\Bbb Q \otimes_{\Bbb Z} \Bbb Q$ respectively.
Does
$$
\phi : \Bbb Q \otimes \Bbb Q \rightarrow \Bbb Q \otimes_{\Bbb Z} \Bbb Q, \;\;\;\;\;\;q \otimes r \mapsto q \otimes \frac{r}{1}
$$
is an isomorphism?
My attempt:
We have a canonical way to identify $\Bbb Q \otimes \Bbb Q $ and $\Bbb Q \otimes_{\Bbb Z} \Bbb Q $. To every element $x \in \Bbb Q \otimes \Bbb Q $ we have $x=x^{(1)} \otimes 1+1 \otimes x^{(2)} $, where $x^{(1)} \in \Bbb Q$ and $x^{(2)} \in \Bbb Q \otimes_{\Bbb Z} \Bbb Q $. This means that every element $x$ of $\Bbb Q \otimes \Bbb Q $ is in $ \Bbb Q \otimes_{\Bbb Z} \Bbb Q $.
Now let us assume that for $x$, $y \in \Bbb Q \otimes \Bbb Q $ holds
$
\phi(x)=\phi(y)
$
i.e. $x^{(1)}=y^{(1)}$ and $x^{(2)}=y^{(2)}$.
So $\frac{y^{(2)}}{1}
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