tensor product $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$ Prove that  $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}=0$ 
I am stuck with this contradiction:
We have $(\mathbb{Z}/n\mathbb{Z}) \mathbb{Q} = \mathbb{Q} $.
and $\mathbb{Q}$ is a $\mathbb{Z}/n\mathbb{Z}$ module and in general for M an A-module we have $A \otimes AM \cong M$ so we should have :
$\mathbb{Q}\otimes \mathbb{Z}/n\mathbb{Z} \cong (\mathbb{Z}/n\mathbb{Z}) \mathbb{Q} \otimes \mathbb{Z}/n\mathbb{Z} \cong \mathbb{Q} $
Could you help?
Thank you.
 A: $\mathbb{Q}$ is not a $\mathbb{Z}/n\mathbb{Z}$-module. A module over $\mathbb{Z}/n\mathbb{Z}$ is the same thing as a $\mathbb{Z}$-module (i.e. an abelian group) in which every element has order a divisor of $n$ (check this!), and the nonzero elements in $\mathbb{Q}$ has infinite order.
With respect to your question, if $\frac{a}{b}\in\mathbb{Q}$ and $\overline{m}\in\mathbb{Z}/n\mathbb{Z}$ then
$$ \frac{a}{b} \otimes \overline{m}=\frac{an}{bn} \otimes \overline{m}=\frac{a}{bn} \otimes n\overline{m}=\frac{a}{bn} \otimes 0=0$$
A: 
We have $(\mathbb{Z}/n\mathbb{Z})\mathbb{Q}=\mathbb{Q}$. 

Really? How do you multiply an element of $\mathbb{Q}$ by an element of $\mathbb{Z}/n\mathbb{Z}$? When $R$ and $R'$ are rings, the notation $RR'$ only makes sense when $R$ and $R'$ are both subrings of some larger ring, or when $R'$ is given the structure of an $R$-algebra or an $R$-module, or vice versa. Which brings us to the next issue. 

$\mathbb{Q}$ is a $\mathbb{Z}/n\mathbb{Z}$ module. 

This is false: $\mathbb{Q}$ cannot be given the structure of a $\mathbb{Z}/n\mathbb{Z}$ module. Indeed, if $\mathbb{Q}$ were a $\mathbb{Z}/n\mathbb{Z}$ module, we would have, for any $a\neq 0$ in $\mathbb{Q}$,
\begin{align*}
\underbrace{(\overline{1} + \dots + \overline{1})}_{n \text{ times}}\cdot a &= \underbrace{\overline{1}\cdot a + \dots + \overline{1}\cdot a}_{n \text{ times}}\\
&= a + \dots + a\\
&= na.
\end{align*}
But the left hand side is $\overline{0}\cdot a = 0 \neq na$.  
Indeed, you can adjust this argument to show that a ring $R$ has the structure of $\mathbb{Z}/n\mathbb{Z}$ module/algebra if and only if the characteristic of $R$ divides $n$. 
