> 
> Is it true that $M \otimes\_\mathbb{Z} N \cong \mathbb{Z}$ (considering $\mathbb{Z}$ as a $\mathbb{Z}$-module) if $M,N$ are cyclic $\mathbb{Z}$-modules (i. e. generated by one element)? 
> 
> 
> 

I would say yes, since $M \otimes\_\mathbb{Z} N$ seems to be cyclic as well while having infinitely many elements, thus being isomorphing to $\mathbb{Z}$ as a group and furthermore as a module.