Show that $\mathbb{Z} \bigotimes \mathbb{Z}=\mathbb{Z}$

Question

I can show that $\mathbb{Z} \bigotimes \mathbb{Z}$ is generated by {$1\bigotimes 1$}. But I could not show that it is linearly independent.

Answer 1

Homomorphisms from $\newcommand{\ot}{\otimes}A\ot B$ to $C$ correspond to bilinear maps from $A\times B$ to $C$. In more detail, if $\Phi:A\times B\to C$ is bilinear, the corresponding homomorphism $\phi$ satisfies $\phi(a\ot b)=\Phi(a,b)$. This is the universal property of the tensor product.

Here, define $\Phi:\newcommand{\Z}{\Bbb Z}\Z\times\Z\to\Z$ by $\Phi(a,b)=ab$. This is bilinear. The corresponding $\phi:\Z\otimes\Z\to\Z$ has $\phi(a\ot b)=ab$. In particular $\phi(n(1\ot 1))=\phi(n\ot 1)=n$. If $n\ne 0$ this means that $n(1\ot1)\ne0$. Thus $\{1\ot 1\}$ is a linearly independent set.