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So, i'm just starting trying to understand tensor products. A basic exercise is given by showing the following equality:

$A \bigotimes \mathbb{Z}_m \cong \frac{A}{mA}$

Well, how about the map $\gamma(a,\bar{x})= ax$ mod $mA$.

And now i'm lost. Can anyone give me some insight please? Thanks!

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    $A \cong A \otimes_{\Bbb{Z}} \Bbb{Z}$ and the kernel of $A \otimes_{\Bbb{Z}} \Bbb{Z} \to A \otimes_{\Bbb{Z}} (\Bbb{Z}/m \Bbb{Z})$ is $A \otimes_{\Bbb{Z}}m \Bbb{Z}=m (A \otimes_{\Bbb{Z}} \Bbb{Z}) \cong mA$. – reuns Aug 18 '19 at 18:52
  • Could you prove $A \cong A \bigotimes_\mathbb{Z} \mathbb{Z}$ for me? –  Aug 18 '19 at 18:54
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    $\otimes_{\Bbb{Z}}$ is a bilinear symbol, in particular $a \otimes_{\Bbb{Z}} n = an \otimes_{\Bbb{Z}} 1$ and the multiplication is defined by $(a \otimes_{\Bbb{Z}} b)(c \otimes_{\Bbb{Z}} d) =ac \otimes_{\Bbb{Z}} bd$ thus $a \mapsto a \otimes_{\Bbb{Z}} 1$ is the isomorphism. – reuns Aug 18 '19 at 18:57

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