Generators of $\mathbb{Z} \times \mathbb{Z}$ Why is $\mathbb{Z} \times \mathbb{Z}$ generated by $(1,0)$ and $(0,1)$? I thought it should be $(1,1)$, but going through the solution of a certain exercise I am working on, it seems I am wrong. 
Can anyone explain this?

Any $(m,n)$ of $\mathbf{Z}\times\mathbf{Z}$ can be written $$(m,n) = (m,0) + (0,n) = \underbrace{(1,0)+\ldots+(1,0)}_{\textrm{$m$ times}} + \underbrace{(0,1)+\ldots+(0,1)}_{\textrm{$n$ times}},$$ showing that $(1,0)$ and $(0,1)$ generate the group $\mathbf{Z}\times\mathbf{Z}$. Summing $k$ times $(1,1)$ will give $(k,k)$, and you'll never be able to get elements of the form $(m,n)$ with $m\not=n$.
