Quotient of $\mathbb{Z} \times \mathbb{Z}$ by subgroup generated by $(n,m)$.

Question

I need to compute this quotient for an application I am working on, but am unsure how to do it.

Answer 1

Write $n=da$ and $m=db$, where $d=gcd(n,m)$ and $a,b$ are coprime. Then $e_1=(a,b)$ can be completed in a basis of $\mathbb{Z}^2$: pick $u,v$ such that $ua+vb=1$ and set $e_2=(-v,u)$.

Then $\mathbb{Z}^2=\mathbb{Z}e_1\oplus \mathbb{Z}e_2$, while the subgroup generated by $(n,m)$ is $\mathbb{Z}de_1$.

Therefore $\mathbb{Z}^2/\mathbb{Z}(n,m)=(\mathbb{Z}e_1\oplus \mathbb{Z}e_2)/\mathbb{Z}de_1\simeq \mathbb{Z}/d\mathbb{Z}\times\mathbb{Z}$, the isomorphism sending the class of $xe_1+ye_2$ to $(\bar{x},y)$.